Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 9 - Probability [Latest edition]

There are four pens: Red, Green, Blue, and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following event.

A: Selecting at least one red pen.

There are four pens: Red, Green, Blue, and Purple in a desk drawer of which two pens are selected at random one after the other with replacement. State the sample space and the following event.

B: Two pens of the same color are not selected.

A coin and a die are tossed simultaneously. Enumerate the sample space and the following event.

A: Getting a Tail and an Odd number

A coin and a die are tossed simultaneously. Enumerate the sample space and the following event.

B: Getting a prime number

A coin and a die are tossed simultaneously. Enumerate the sample space and the following event.

C: Getting a head and a perfect square.

Find n(S) of the following random experiment.

From an urn containing 5 gold and 3 silver coins, 3 coins are drawn at random

Find n(S) of the following random experiment.

5 letters are to be placed into 5 envelopes such that no envelope is empty.

Find n(S) of the following random experiment.

6 books of different subjects arranged on a shelf.

Find n(S) of the following random experiment.

3 tickets are drawn from a box containing 20 lottery tickets.

Two fair dice are thrown. State the sample space and write favorable outcomes for the following event:

A: Sum of numbers on two dice is divisible by 3 or 4

Two fair dice are thrown. State the sample space and write favorable outcomes for the following event:

B: Sum of numbers on two dice is 7

Two fair dice are thrown. State the sample space and write favorable outcomes for the following event:

C: Odd number on the first die.

Two fair dice are thrown. State the sample space and write favorable outcomes for the following event:

D: Even number on the first die.

Two fair dice are thrown. State the sample space and write favorable outcomes for the following events:

Check whether events A and B are mutually exclusive and exhaustive.

A: Sum of numbers on two dice is divisible by 3 or 4.

B: Sum of numbers on two dice is 7.

Two fair dice are thrown. State the sample space and write favorable outcomes for the following events:

Check whether events C and D are mutually exclusive and exhaustive

C: Odd number on the first die.

D: Even number on the first die.

A bag contains four cards marked as 5, 6, 7, and 8. Find the sample space if two cards are drawn at random with replacement.

A bag contains four cards marked as 5, 6, 7, and 8. Find the sample space if two cards are drawn at random without replacement.

A fair die is thrown two times. Find the probability that sum of the numbers on them is 5

A fair die is thrown two times. Find the probability that sum of the numbers on them is at least 8

A fair die is thrown two times. Find the probability that the first throw gives a multiple of 2 and the second throw gives a multiple of 3.

A fair die is thrown two times. Find the probability that the product of numbers on them is 12.

Two cards are drawn from a pack of 52 cards. Find the probability that one is a face card and the other is an ace card

Two cards are drawn from a pack of 52 cards. Find the probability that one is club and the other is a diamond.

Two cards are drawn from a pack of 52 cards. Find the probability that both are from the same suit.

Two cards are drawn from a pack of 52 cards. Find the probability that both are red cards

Two cards are drawn from a pack of 52 cards. Find the probability that one is a heart card and the other is a non-heart card

Three cards are drawn from a pack of 52 cards. Find the chance that two are queen cards and one is an ace card

Three cards are drawn from a pack of 52 cards. Find the chance that at least one is a diamond card

Three cards are drawn from a pack of 52 cards. Find the chance that all are from the same suit

Three cards are drawn from a pack of 52 cards. Find the chance that they are a king, a queen, and a jack

From a bag containing 10 red, 4 blue and 6 black balls, a ball is drawn at random. Find the probability of drawing a red ball.

From a bag containing 10 red, 4 blue and 6 black balls, a ball is drawn at random. Find the probability of drawing a blue or black ball

From a bag containing 10 red, 4 blue and 6 black balls, a ball is drawn at random. Find the probability of drawing not a black ball.

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that, Number on the ticket is divisible by 6

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that, Number on the ticket is a perfect square

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that, Number on the ticket is prime

A box contains 75 tickets numbered 1 to 75. A ticket is drawn at random from the box. Find the probability that, Number on the ticket is divisible by 3 and 5

What is the chance that a leap year, selected at random, will contain 53 Sundays?

Find the probability of getting both red balls, when from a bag containing 5 red and 4 black balls, two balls are drawn, with replacement

Find the probability of getting both red balls, when from a bag containing 5 red and 4 black balls, two balls are drawn, without replacement

A room has three sockets for lamps. From a collection 10 bulbs of which 6 are defective. At night a person selects 3 bulbs, at random and puts them in sockets. What is the probability that room is still dark

A room has three sockets for lamps. From a collection 10 bulbs of which 6 are defective. At night a person selects 3 bulbs, at random and puts them in sockets. What is the probability that the room is lit

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement vowels are always together.

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement vowels are never together

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement O is at the begining and end with T

Letters of the word MOTHER are arranged at random. Find the probability that in the arrangement starting with a vowel and end with a consonant

4 letters are to be posted in 4 post boxes. If any number of letters can be posted in any of the 4 post boxes, what is the probability that each box contains only one letter?

15 professors have been invited for a round table conference by Vice chancellor of a university. What is the probability that two particular professors occupy the seats on either side of the Vice chancellor during the conference

A bag contains 7 black and 4 red balls. If 3 balls are drawn at random find the probability that all are black

A bag contains 7 black and 4 red balls. If 3 balls are drawn at random find the probability that one is black and two are red.

First 6 faced die which is numbered 1 through 6 is thrown then a 5 faced die which is numbered 1 through 5 is thrown. What is the probability that the sum of the numbers on the upper faces of the dice is divisible by 2 or 3?

A card is drawn from a pack of 52 cards. What is the probability that, card is either red or black?

A card is drawn from a pack of 52 cards. What is the probability that, card is either black or a face card?

A girl is preparing for National Level Entrance exam and State Level Entrance exam for professional courses. The chances of her cracking National Level exam is 0.42 and that of State Level exam is 0.54. The probability that she clears both the exams is 0.11. Find the probability that

1. She cracks at least one of the two exams
2. She cracks only one of the two
3. She cracks none.

A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that, number on the ticket is a perfect square or divisible by 4

A bag contains 75 tickets numbered from 1 to 75. One ticket is drawn at random. Find the probability that, number on the ticket is a prime number or greater than 40

The probability that a student will pass in French is 0.64, will pass in Sociology is 0.45, and will pass in both is 0.40. What is the probability that the student will pass in at least one of the two subjects?

Two fair dice are thrown. Find the probability that number on the upper face of the first die is 3 or sum of the numbers on their upper faces is 6

For two events A and B of a sample space S, if P(A) =` 3/8`, P(B) = `1/2` and P(A ∪ B) = `5/8`. Find the value of the following: P(A ∩ B)

For two events A and B of a sample space S, if P(A) = `3/8` , P(B) = `1/2` and P(A ∪ B) = `5/8` . Find the value of the following: P(A' ∩ B')

For two events A and B of a sample space S, if P(A) =` 3/8`, P(B) = `1/2` and P(A ∪ B) = `5/8`. Find the value of the following: P(A' ∪ B')

For two events A and B of a sample space S, if P(A ∪ B) = `5/6`, P(A ∩ B) = `1/3` and P(B') = `1/3`, then find P(A).

A bag contains 5 red, 4 blue and an unknown number m of green balls. If the probability of getting both the balls green, when two balls are selected at random is `1/7`, find m

Form a group of 4 men, 4 women and 3 children, 4 persons are selected at random. Find the probability that, no child is selected.

Form a group of 4 men, 4 women and 3 children, 4 persons are selected at random. Find the probability that, exactly 2 men are selected

A number is drawn at random from the numbers 1 to 50. Find the probability that it is divisible by 2 or 3 or 10

A bag contains 3 red marbles and 4 blue marbles. Two marbles are drawn at random without replacement. If the first marble drawn is red, what is the probability the second marble is blue?

A box contains 5 green pencils and 7 yellow pencils. Two pencils are chosen at random from the box without replacement. What is the probability that both are yellow?

In a sample of 40 vehicles, 18 are red, 6 are trucks, of which 2 are red. Suppose that a randomly selected vehicle is red. What is the probability it is a truck?

From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when first card drawn is kept aside

From a pack of well-shuffled cards, two cards are drawn at random. Find the probability that both the cards are diamonds when the first card drawn is replaced in the pack

A, B, and C try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are `3/4, 1/2` and `5/8`. Find the probability that the target

1. is hit exactly by one of them
2. is not hit by any one of them
3. is hit
4. is exactly hit by two of them

The probability that a student X solves a problem in dynamics is `2/5` and the probability that student Y solves the same problem is `1/4`. What is the probability that

1. the problem is not solved
2. the problem is solved
3. the problem is solved exactly by one of them

A speaks truth in 80% of the cases and B speaks truth in 60% of the cases. Find the probability that they contradict each other in narrating an incident

Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery

The follwoing table summarizes their response:

If one person from the 200 patients is selected at random, determine the probability that the person was satisfied given that the person had Throat surgery.

Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery

The follwoing table summarizes their response:

If one person from the 200 patients is selected at random, determine the probability that person was unsatisfied given that the person had eye surgery

Two hundred patients who had either Eye surgery or Throat surgery were asked whether they were satisfied or unsatisfied regarding the result of their surgery.

The following table summarizes their response:

If one person from the 200 patients is selected at random, determine the probability the person had Throat surgery given that the person was unsatisfied.

Two dice are thrown together. Let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent?

The probability that a man who is 45 years old will be alive till he becomes 70 is `5/12`. The probability that his wife who is 40 years old will be alive till she becomes 65 is `3/8`. What is the probability that, 25 years hence,

1. the couple will be alive
2. exactly one of them will be alive
3. none of them will be alive
4. at least one of them will be alive

A box contains 10 red balls and 15 green balls. Two balls are drawn in succession without replacement. What is the probability that, the first is red and the second is green?

A box contains 10 red balls and 15 green balls. Two balls are drawn in succession without replacement. What is the probability that, one is red and the other is green?

A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the probability that, both the balls are of the same color?

A bag contains 3 yellow and 5 brown balls. Another bag contains 4 yellow and 6 brown balls. If one ball is drawn from each bag, what is the probability that, the balls are of different color?

An urn contains 4 black, 5 white and 6 red balls. Two balls are drawn one after the other without replacement. What is the probability that at least one of them is black?

Three fair coins are tossed. What is the probability of getting three heads given that at least two coins show heads?

Two cards are drawn one after the other from a pack of 52 cards without replacement. What is the probability that both the cards drawn are face cards?

Bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag, and then a ball is drawn from that bag. Find the probability that both the balls drawn are of same color

A bag contains 3 red and 5 white balls. Two balls are drawn at random one after the other without replacement. Find the probability that both the balls are white.

Solution: Let,

A : First ball drawn is white

B : second ball drawn in white.

P(A) = `square/square`

After drawing the first ball, without replacing it into the bag a second ball is drawn from the remaining `square` balls.

∴ P(B/A) = `square/square`

∴ P(Both balls are white) = P(A ∩ B)

`= "P"(square) * "P"(square)`

`= square * square`

= `square`

A family has two children. Find the probability that both the children are girls, given that atleast one of them is a girl.

There are three bags, each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles and Bag 3 has 45 red and 55 blue marbles. One of the bags is chosen at random and a marble is picked from the chosen bag. What is the probability that the chosen marble is red?

A box contains 2 blue and 3 pink balls and another box contains 4 blue and 5 pink balls. One ball is drawn at random from one of the two boxes and it is found to be pink. Find the probability that it was drawn from first box

A box contains 2 blue and 3 pink balls and another box contains 4 blue and 5 pink balls. One ball is drawn at random from one of the two boxes and it is found to be pink. Find the probability that it was drawn from second box

There is a working women's hostel in a town, where 75% are from neighbouring town. The rest all are from the same town. 48% of women who hail from the same town are graduates and 83% of the women who have come from the neighboring town are also graduates. Find the probability that a woman selected at random is a graduate from the same town

If E₁ and E₂ are equally likely, mutually exclusive and exhaustive events and `"P"("A"/"E"_1 )` = 0.2, `"P"("A"/"E"_2)` = 0.3. Find `"P"("E"_1/"A")`

Jar I contains 5 white and 7 black balls. Jar II contains 3 white and 12 black balls. A fair coin is flipped; if it is Head, a ball is drawn from Jar I, and if it is Tail, a ball is drawn from Jar II. Suppose that this experiment is done and a white ball was drawn. What is the probability that this ball was in fact taken from Jar II?

A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive result when applied to a non-sufferer. It is estimated that 0.5% of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that: given a positive result, the person is a sufferer 

A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive result when applied to a non-sufferer. It is estimated that 0.5% of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the probability that: given a negative result, the person is a non-sufferer

A doctor is called to see a sick child. The doctor has prior information that 80% of the sick children in that area have the flu, while the other 20% are sick with measles. Assume that there is no other disease in that area. A well-known symptom of measles is rash. From the past records, it is known that, chances of having rashes given that sick child is suffering from measles is 0.95. However occasionally children with flu also develop rash, whose chance are 0.08. Upon examining the child, the doctor finds a rash. What is the probability that child is suffering from measles?

2% of the population have a certain blood disease of a serious form: 10% have it in a mild form; and 88% don't have it at all. A new blood test is developed; the probability of testing positive is `9/10` if the subject has the serious form, `6/10` if the subject has the mild form, and `1/10` if the subject doesn't have the disease. A subject is tested positive. What is the probability that the subject has serious form of the disease?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. What is the probability that it lands head up?

A box contains three coins: two fair coins and one fake two-headed coin is picked randomly from the box and tossed. If happens to be head, what is the probability that it is the two-headed coin?

There are three social media groups on a mobile: Group I, Group II and Group III. The probabilities that Group I, Group II and Group III sending the messages on sports are `2/5, 1/2`, and `2/3` respectively. The probability of opening the messages by Group I, Group II and Group III are `1/2, 1/4` and `1/4` respectively. Randomly one of the messages is opened and found a message on sports. What is the probability that the message was from Group III

(Activity):

Mr. X goes to office by Auto, Car, and train. The probabilities him travelling by these modes are `2/7, 3/7, 2/7` respectively. The chances of him being late to the office are `1/2, 1/4, 1/4` respectively by Auto, Car, and train. On one particular day, he was late to the office. Find the probability that he travelled by car.

Solution: Let A, C and T be the events that Mr. X goes to office by Auto, Car and Train respectively. Let L be event that he is late.

Given that P(A) = `square`, P(C) = `square`

P(T) = `square`

P(L/A) = `1/2`, P(L/C) = `square` P(L/T) = `1/4`

P(L) = P(A ∩ L) + P(C ∩ L) + P(T ∩ L)

`="P"("A")*"P"("L"//"A") + "P"("C")*"P"("L"//"C") + "P"("T")*"P"("L"//"T")`

`= square * square + square * square + square * square`

`= square + square + square`

`= square`

`"P"("C"//"L") = ("P"("L" ∩ "C"))/("P"("L"))`

= `("P"("C") * "P"("L"//"C"))/("P"("L"))`

`= (square * square)/square`

`= square`

If odds in favour of X solving a problem are 4:3 and odds against Y solving the same problem are 2:3. Find probability of: X solving the problem

If odds in favour of X solving a problem are 4:3 and odds against Y solving the same problem are 2:3. Find probability of: Y solving the problem

The odds against John solving a problem are 4 to 3 and the odds in favor of Rafi solving the same problem are 7 to 5. What is the chance that the problem is solved when both them try it?

The odds against student X solving a statistics problem are 8:6 and odds in favour of student y solving the same problem are 14:16. Find is the chance that the problem will be solved if they try it independently

The odds against student X solving a statistics problem are 8:6 and odds in favour of student y solving the same problem are 14:16. Find is the chance that neither of them solves the problem

The odds against a husband who is 60 years old, living till he is 85 are 7:5. The odds against his wife who is now 56, living till she is 81 are 5:3. Find the probability that at least one of them will be alive 25 years hence

The odds against a husband who is 60 years old, living till he is 85 are 7:5. The odds against his wife who is now 56, living till she is 81 are 5:3. Find the probability that exactly one of them will be alive 25 years hence

There are three events A, B and C, one of which must, and only one can happen. The odds against the event A are 7:4 and odds against event B are 5:3. Find the odds against event C

In a single toss of a fair die, what are the odds against the event that number 3 or 4 turns up?

The odds in favour of A winning a game of chess against B are 3:2. If three games are to be played, what are the odds in favour of A's winning at least two games out of the three?

Select the correct option from the given alternatives :

There are 5 girls and 2 boys, then the probability that no two boys are sitting together for a photograph is

Select the correct option from the given alternatives :

In a jar there are 5 black marbles and 3 green marbles. Two marbles are picked randomly one after the other without replacement. What is the possibility that both the marbles are black?

Select the correct option from the given alternatives :

Two dice are thrown simultaneously. Then the probability of getting two numbers whose product is even is

Select the correct option from the given alternatives :

In a set of 30 shirts, 17 are white and rest are black. 4 white and 5 black shirts are tagged as ‘PARTY WEAR’. If a shirt is chosen at random from this set, the possibility of choosing a black shirt or a ‘PARTY WEAR’ shirt is

Select the correct option from the given alternatives :

There are 2 shelves. One shelf has 5 Physics and 3 Biology books and the other has 4 Physics and 2 Biology books. The probability of drawing a Physics book is

Select the correct option from the given alternatives :

Two friends A and B apply for a job in the same company. The chances of A getting selected is `2/5` and that of B is `4/7`. The probability that both of them get selected is

Select the correct option from the given alternatives :

The probability that a student knows the correct answer to a multiple choice question is `2/3`. If the student does not know the answer, then the student guesses the answer. The probability of the guessed answer being correct is `1/4`. Given that the student has answered the question correctly, the probability that the student knows the correct answer is

Select the correct option from the given alternatives :

Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. The probability that it was drawn from Bag II

Select the correct option from the given alternatives :

A fair is tossed twice. What are the odds in favour of getting 4, 5 or 6 on the first toss and 1, 2, 3 or 4 on the second die?

Select the correct option from the given alternatives :

The odds against an event are 5:3 and the odds in favour of another independent event are 7:5. The probability that at least one of the two events will occur is

Solve the following:

The letters of the word 'EQUATION' are arranged in a row. Find the probability that All the vowels are together

Solve the following:

The letters of the word 'EQUATION' are arranged in a row. Find the probability that Arrangement starts with a vowel and ends with a consonant

Solve the following:

There are 6 positive and 8 negative numbers. Four numbers are chosen at random, without replacement, and multiplied. Find the probability that the product is a positive number.

Solve the following:

Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is and even number?

Solve the following:

If P(A ∩ B) = `1/2`, P(B ∩ C) = `1/3`, P(C ∩ A) = `1/6` then find P(A), P(B) and P(C), If A,B,C are independent events.

Solve the following:

If the letters of the word 'REGULATIONS' be arranged at random, what is the probability that there will be exactly 4 letters between R and E?

Solve the following:

In how many ways can the letters of the word ARRANGEMENTS be arranged? Find the chance that an arrangement chosen at random begins with the letters EE.

Solve the following:

In how many ways can the letters of the word ARRANGEMENTS be arranged? Find the probability that the consonants are together

Solve the following:

A letter is taken at random from the letters of the word 'ASSISTANT' and another letter is taken at random from the letters of the word 'STATISTICS'. Find the probability that the selected letters are the same

Solve the following:

A die is loaded in such a way that the probability of the face with j dots turning up is proportional to j for j = 1, 2, .......6. What is the probability, in one roll of the die, that an odd number of dots will turn up?

Solve the following:

An urn contains 5 red balls and 2 green balls. A ball is drawn. If its green, a red ball is added to the urn and if its red, a green ball is added to the urn. (The original ball is not returned to the urn). Then a second ball is drawn. What is the probability that the second ball is red?

Solve the following:

The odds against A solving a certain problem are 4 to 3 and the odds in favor of solving the same problem are 7 to 5 find the probability that the problem will be solved

Solve the following:

If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("A'"/"B")`

Solve the following:

If P(A) = `"P"("A"/"B") = 1/5, "P"("B"/"A") = 1/3` the find `"P"("B'"/"A'")`

Solve the following:

Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find P(B)

Solve the following:

Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find `"P"("A"/"B")`

Solve the following:

Let A and B be independent events with P(A) = `1/4`, and P(A ∪ B) = 2P(B) – P(A). Find `"P"("B'"/"A")`

Solve the following:

Find the probability that a year selected will have 53 Wednesdays

Solve the following:

The chances of P, Q and R, getting selected as principal of a college are `2/5, 2/5, 1/5` respectively. Their chances of introducing IT in the college are `1/2, 1/3, 1/4` respectively. Find the probability that IT is introduced in the college after one of them is selected as a principal

Solve the following:

The chances of P, Q and R, getting selected as principal of a college are `2/5, 2/5, 1/5` respectively. Their chances of introducing IT in the college are `1/2, 1/3, 1/4` respectively. Find the probability that IT is introduced by Q

Solve the following:

Suppose that five good fuses and two defective ones have been mixed up. To find the defective fuses, we test them one-byone, at random and without replacement. What is the probability that we are lucky and fine both of the defective fuses in the first two tests?

Solve the following:

For three events A, B and C, we know that A and C are independent, B and C are independent, A and B are disjoint, P(A ∪ C) = `2/3`, P(B ∪ C) = `3/4`, P(A ∪ B ∪ C) = `11/12`. Find P(A), P(B) and P(C)

Solve the following:

The ratio of Boys to Girls in a college is 3:2 and 3 girls out of 500 and 2 boys out of 50 of that college are good singers. A good singer is chosen what is the probability that the chosen singer is a girl?

Solve the following:

A and B throw a die alternatively till one of them gets a 3 and wins the game. Find the respective probabilities of winning. (Assuming A begins the game)

Solve the following:

Consider independent trails consisting of rolling a pair of fair dice, over and over What is the probability that a sum of 5 appears before sum of 7?

Solve the following:

A machine produces parts that are either good (90%), slightly defective (2%), or obviously defective (8%). Produced parts get passed through an automatic inspection machine, which is able to detect any part that is obviously defective and discard it. What is the quality of the parts that make it throught the inspection machine and get shipped?

Solve the following:

Given three identical boxes, I, II, and III, each containing two coins. In box I, both coins are gold coins, in box II, both are silver coins and in box III, there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, what is the probability that the other coin in the box is also of gold?

Solve the following:

In a factory which manufactures bulbs, machines A, B and C manufacture respectively 25%, 35% and 40% of the bulbs. Of their outputs, 5, 4 and 2 percent are respectively defective bulbs. A bulbs is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine B?

Solve the following:

A family has two children. One of them is chosen at random and found that the child is a girl. Find the probability that both the children are girls

Solve the following:

A family has two children. One of them is chosen at random and found that the child is a girl. Find the probability that both the children are girls given that at least one of them is a girl