Advertisements
Advertisements
Question
A couple has two children. Find the probability that both the children are (i) males, if it is known that at least one of the children is male. (ii) females, if it is known that the elder child is a female.
Solution
Consider the given events.
A = Both the children are female.
B = The elder child is a female.
C = At least one child is a male.
D = Both children are male.
\[\text{ Clearly } , \]
\[S = \left\{ M_1 M_2 , M_1 F_2 , F_1 M_2 , F_1 F_2 \right\}\]
\[A = \left\{ F_1 F_2 \right\}\]
\[B = \left\{ F_1 M_2 , F_1 F_2 \right\}\]
\[C = \left\{ M_1 F_2 , F_1 M_2 , M_1 M_2 \right\} \]
\[D = \left\{ M_1 M_2 \right\}\]
[Here, first child is elder and second is younger]
\[D \cap C = \left\{ M_1 M_2 \right\} \text{ and } A \cap B = \left\{ F_1 F_2 \right\}\]
\[\left( i \right) \text{ Required probability } = P\left( D/C \right) = \frac{n\left( D \cap C \right)}{n\left( C \right)} = \frac{1}{3}\]
\[\left( ii \right) \text{ Required probability } = P\left( A/B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{1}{2}\]
APPEARS IN
RELATED QUESTIONS
In a shop X, 30 tins of pure ghee and 40 tins of adulterated ghee which look alike, are kept for sale while in shop Y, similar 50 tins of pure ghee and 60 tins of adulterated ghee are there. One tin of ghee is purchased from one of the randomly selected shops and is found to be adulterated. Find the probability that it is purchased from shop Y. What measures should be taken to stop adulteration?
A card is drawn from a well-shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.
Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and third card drawn is an ace?
If P (A) = 0.4, P (B) = 0.8, P (B/A) = 0.6. Find P (A/B) and P (A ∪ B).
A die is rolled. If the outcome is an odd number, what is the probability that it is prime?
A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find P (E/F). If F is the event "5 appears on at least one die", find P (E/F).
A coin is tossed three times. Let the events A, B and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.
Check the independence of A and B.
Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find P (A ∩ B).
Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find \[P \overline A \cup \overline B \] .
If P (not B) = 0.65, P (A ∪ B) = 0.85, and A and B are independent events, then find P (A).
If A and B are two independent events such that P (`bar A` ∩ B) = 2/15 and P (A ∩`bar B` ) = 1/6, then find P (B).
Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem is 3/5. Find the probability that none of the two will be able to solve the problem.
The probabilities of two students A and B coming to the school in time are \[\frac{3}{7}\text { and }\frac{5}{7}\] respectively. Assuming that the events, 'A coming in time' and 'B coming in time' are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that one of them is black and other is red.
Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.
A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black?
Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4.
Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.
A, B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?
A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.
One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.
If A and B are independent events, then write expression for P(exactly one of A, B occurs).
If one ball is drawn at random from each of three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, then the probability that 2 white and 1 black balls will be drawn is
Three faces of an ordinary dice are yellow, two faces are red and one face is blue. The dice is rolled 3 times. The probability that yellow red and blue face appear in the first second and third throws respectively, is
The probability that a leap year will have 53 Fridays or 53 Saturdays is
A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is
An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is
If S is the sample space and P (A) = \[\frac{1}{3}\]P (B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) =
If A and B are two events, then P (`overline A` ∩ B) =
If P (A ∪ B) = 0.8 and P (A ∩ B) = 0.3, then P \[\left( A \right)\] \[\left( A \right)\] + P \[\left( B \right)\] =
A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, the probability chosen to be white is
Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both dice will be awarded the prize. If A throws first, then the probability that Bwins the game is
Choose the correct alternative in the following question: \[\text{ Let } P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13} . \text{ Then } , P\left( \overline{ A }|B \right) = \]
Mark the correct alternative in the following question:
\[\text{ If } P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}, \text{ then } P\left( B|\overline{ A } \right) = \]
Mark the correct alternative in the following question:
\[\text{ If A and B are two events such that} P\left( A \right) \neq 0 \text{ and } P\left( B \right) \neq 1,\text{ then } P\left( \overline{ A }|\overline{ B }\right) = \]
Mark the correct alternative in the following question:
If two events are independent, then
Mark the correct alternative in the following question:
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is
A, B and C throw a pair of dice in that order alternatively till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first.
