Advertisements
Advertisements
Question
A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} Find P ((E ∪ F)|G) and P ((E ∩ G)|G)
Solution
E = {1, 3, 5}, F = {2, 3}, G = {2, 3, 4, 5}
E ∩ G = {3, 5}, F ∩ G = {2, 3}, (E ∩ F) ∩ G = {3}
P(E ∩ G) = `2/6`, P(F ∩ G) = `2/6`, P[(E ∩ F) ∩ G] = `1/6`
Now P(E ∪ F|G) = P(E|G) + P(F|G) − P[(E ∩ F) ∩ G]
=`(P(E ∩ G))/(P(G)) + (P(E ∩ G))/(P(G)) - (P[(E ∩ F) ∩ G])/(P(G))`
= `(2/6 ÷ 4/6) + (2/6 ÷ 4/6) - (1/6 ÷ 4/6)`
= `2/4 + 2/4 - 1/4`
= `3/4`
P(E ∩ F|G) = `(P[(E ∩ F) ∩ G])/(P(G))`
= `1/6 ÷ 4/6`
= `1/4`
APPEARS IN
RELATED QUESTIONS
A fair coin is tossed five times. Find the probability that it shows exactly three times head.
The probability that a certain kind of component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 tested components survive
A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y.
Suppose that 80% of all families own a television set. If 5 families are interviewed at random, find the probability that
a. three families own a television set.
b. at least two families own a television set.
Evaluate P(A ∪ B), if 2P(A) = P(B) = `5/13` and P(A | B) = `2/5`
Determine P(E|F).
A coin is tossed three times, where
E: at least two heads, F: at most two heads
Determine P(E|F).
Two coins are tossed once, where
E: no tail appears, F: no head appears
Determine P(E|F).
A die is thrown three times,
E: 4 appears on the third toss, F: 6 and 5 appears respectively on first two tosses
A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} Find P (E|G) and P (G|E)
An instructor has a question bank consisting of 300 easy True/False questions, 200 difficult True/False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple-choice question?
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
- both balls are red.
- first ball is black and second is red.
- one of them is black and other is red.
If A and B are events such as that P(A) = `1/2`, P(B) = `1/3` and P(A ∩ B) = `1/4`, then find
1) P(A / B)
2) P(B / A)
Five dice are thrown simultaneously. If the occurrence of an odd number in a single dice is considered a success, find the probability of maximum three successes.
A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.
Three cards are drawn at random (without replacement) from a well-shuffled pack of 52 playing cards. Find the probability distribution of the number of red cards. Hence, find the mean of the distribution.
In a college, 70% of students pass in Physics, 75% pass in Mathematics and 10% of students fail in both. One student is chosen at random. What is the probability that:
(i) He passes in Physics and Mathematics?
(ii) He passes in Mathematics given that he passes in Physics.
(iii) He passes in Physics given that he passes in Mathematics.
A pair of dice is thrown. If sum of the numbers is an even number, what is the probability that it is a perfect square?
In an examination, 30% of students have failed in subject I, 20% of the students have failed in subject II and 10% have failed in both subject I and subject II. A student is selected at random, what is the probability that the student has failed in at least one subject?
A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that, one is white and other is black?
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 ball is black?
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
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
Can two events be mutually exclusive and independent simultaneously?
A problem in Mathematics is given to three students whose chances of solving it are `1/3, 1/4` and `1/5`. What is the probability that the problem is solved?
A problem in Mathematics is given to three students whose chances of solving it are `1/3, 1/4` and `1/5`. What is the probability that exactly one of them will solve it?
One bag contains 5 white and 3 black balls. Another bag contains 4 white and 6 black balls. If one ball is drawn from each bag, find the probability that both are white
Given P(A) = 0.4 and P(A ∪ B) = 0.7 Find P(B) if A and B are mutually exclusive
Given P(A) = 0.4 and P(A ∪ B) = 0.7 Find P(B) if A and B are independent events
A year is selected at random. What is the probability that it is a leap year which contains 53 Sundays
In a multiple-choice question, there are three options out of which only one is correct. A person is guessing the answer at random. If there are 7 such questions, then the probability that he will get exactly 4 correct answers is ______
If X denotes the number of ones in five consecutive throws of a dice, then P(X = 4) is ______
Two dice are thrown. Find the probability that the sum of numbers appearing is more than 11, is ______.
Let A and B be two events. If P(A) = 0.2, P(B) = 0.4, P(A ∪ B) = 0.6, then P(A|B) is equal to ______.
If two balls are drawn from a bag containing 3 white, 4 black and 5 red balls. Then, the probability that the drawn balls are of different colours is:
Let A and B be two non-null events such that A ⊂ B. Then, which of the following statements is always correct?
If A and B are two events such that P(A) = `1/3`, P(B) = `1/5` and P(A ∪ B) = `1/2`, then P(A|B') + P(B|A') is equal to ______.
If for any two events A and B, P(A) = `4/5` and P(A ∩ B) = `7/10`, then `P(B/A)` is equal to ______.
