Advertisements
Advertisements
Question
If A and B are two events such that 2 P (A) = P (B) = \[\frac{5}{13}\] and P (A/B) = \[\frac{2}{5},\] find P (A ∪ B).
Solution
\[\text{ Given } : \]
\[2P\left( A \right) = P\left( B \right) = \frac{5}{13} \]
\[P\left( A/B \right) = \frac{2}{5}\]
\[ \therefore P\left( A \right) = \frac{5}{26} \]
\[ P\left( B \right) = \frac{5}{13}\]
\[\text{ Now } , \]
\[ P\left( A/B \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)}\]
\[ \Rightarrow \frac{2}{5} = \frac{P\left( A \cap B \right)}{\frac{5}{13}}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{2}{5} \times \frac{5}{13} = \frac{2}{13}\]
\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{5}{26} + \frac{5}{13} - \frac{2}{13}\]
\[ = \frac{11}{26}\]
APPEARS IN
RELATED QUESTIONS
A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins
Compute P (A/B), if P (B) = 0.5 and P (A ∩ B) = 0.32
If P (A) = 0.4, P (B) = 0.3 and P (B/A) = 0.5, find P (A ∩ B) and P (A/B).
A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.
A bag contains 20 tickets, numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and the second an odd number.
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 A and B are two events such that \[ P\left( A \right) = \frac{1}{3}, P\left( B \right) = \frac{1}{4} \text{ and } P\left( A \cup B \right) = \frac{5}{12}, \text{ then find } P\left( A|B \right) \text{ and } P\left( B|A \right) . \]
A die is thrown three times. Find P (A/B) and P (B/A), if
A = 4 appears on the third toss, B = 6 and 5 appear respectively on first two tosses.
A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that
(i) the youngest is a girl (b) at least one is a girl.
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.
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. C and A
If A and B are two independent events such that P (A ∪ B) = 0.60 and P (A) = 0.2, find P(B).
The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that (i) at least one of the events will occur, and (ii) none of the events will occur.
Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are: `1 - (1 - p_1 )(1 -p_2 ) `
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that first ball is black and second is red.
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?
Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.
A bag contains 7 white, 5 black and 4 red balls. Four balls are drawn without replacement. Find the probability that at least three balls are black.
In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.
The contents of three bags I, II and III are as follows:
Bag I : 1 white, 2 black and 3 red balls,
Bag II : 2 white, 1 black and 1 red ball;
Bag III : 4 white, 5 black and 3 red balls.
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?
An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.
A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.
If P (A) = 0.3, P (B) = 0.6, P (B/A) = 0.5, find P (A ∪ B).
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
A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit 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
If A and B are two events, then P (`overline A` ∩ B) =
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
Mark the correct alternative in the following question:
If A and B are two events such that P(A) = \[\frac{4}{5}\] , and \[P\left( A \cap B \right) = \frac{7}{10}\] , then P(B|A) =
Choose the correct alternative in the following question:
\[\text{ If } P\left( A \right) = \frac{2}{5}, P\left( B \right) = \frac{3}{10} \text{ and } P\left( A \cap B \right) = \frac{1}{5}, \text{ then } , P\left( \overline { A }|\overline{ B } \right) P\left( \overline{ B }|\overline{ A } \right) \text{ is equal to } \]
Mark the correct alternative in the following question:
\[\text{ If} P\left( A \right) = 0 . 4, P\left( B \right) = 0 . 8 \text{ and } P\left( B|A \right) = 0 . 6, \text{ then } P\left( A \cup B \right) = \]
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:A flash light has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then the probability that both are dead is
Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is
Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is
Mark the correct alternative in the following question:
\[\text{ If A and B are two events such that } P\left( A|B \right) = p, P\left( A \right) = p, P\left( B \right) = \frac{1}{3} \text{ and } P\left( A \cup B \right) = \frac{5}{9}, \text{ then} p = \]
Mark the correct alternative in the following question:
\[\text{ Let A and B be two events . If } P\left( A \right) = 0 . 2, P\left( B \right) = 0 . 4, P\left( A \cup B \right) = 0 . 6, \text{ then } P\left( A|B \right) \text{ is equal to} \]
