Math/Stat 304
WS III
Conditional Probability
1. Suppose that a fair die is thrown once. Find the probability that the outcome is 6 given that the
outcome is even.
2. Toss a fair coin 3 successive times. Find the conditional probability P(A|B) where
A = “more heads than tails”, and B = “first toss is a head”.
3. A fair 4-sided die is rolled twice and we assume all sixteen possible outcomes are equiprobable.
Let A =”max of 1st and 2nd = m”, and B= “min of 1st and 2nd toss is 3}. (Here m = 1, 2, 3, 4.) Find
P(A|B) for each m.
4.
A conservative design team, call it C, and an innovative design team, call it N, are asked to
separately design a new product within a month. From past experience, we know:
(a) The probability that team C is successful is 2/3.
(b) The probability that team N is successful is ½.
(c) The probability that at least one team is succssful is ¾.
Assuming that exactly one successful design is produced, what it the probability that it was
designed by team N?
5.
Three cards are drawn from a deck of 52. Find the probability that there is a pair in the hand (but
not 3 of a kind) given that at least one ace and at least one king are in the hand.
6. Albertine asks her neighbor, Marcel, to water a sickly African violet while she travels to Nice for
spring break. Without water, the plant will die with probability 0.8; with water, it will die with
probability 0.15. Albertine is 90% certain that Marcel will remember to water the plant. What is the
probability that the African violet will be alive when Albertine returns from Nice?
7. A fair coin is flipped 25 times.
(a) Find the probability that exactly three heads appear.
(b) Find the conditional probability of exactly 3 heads appearing given that at least one head
appears.
The following exercises are from Sheldon Ross’ A First Course in Probability.
8. Celine is undecided as to whether to take a math course or a chemistry course. She estimates that
her probability of receiving a grade of A would be ½ in a math course and 1/3 in a chemistry
course. If Celine decides to base her decision on the flip of a fair coin, what is the probability that
she gets an A in chemistry?
9. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that
the dice land on different numbers?
10. If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that
the sum of the dice is i? Compute for all values of i between 2 and 12.
2
11. What is the probability that at least one of a pair of fair dice lands on 6, given that the sum of the
dice is i, where i = 2, 3, ... , 12?
12.. An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without
replacement, what is the probability that the first 2 selected are white and the last 2 black?
13. Consider an urn containing 12 balls, of which 8 are white. A sample of size 4 is to be drawn with
replacement (without replacement). What is the conditional probability (in each case) that the first
and third balls drawn will be white given that the sample drawn contains exactly 3 white balls?
14. A couple has 2 children. What is the probability that both are girls if the older of the two is a girl?
15. Consider 3 urns. Urn A contains 2 white and 4 red balls, urn B contains 8 white and 4 red balls,
and urn C contains 1 white and 3 red balls. If one ball is selected from each urn, what is the
probability that the ball chosen from urn A was white given that exactly 2 white balls were
selected?
16. Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing
cards. Compute the conditional probability that the second and third cards are spades given the first
card selected is a spade.
17. Two cards are randomly chosen without replacement from an ordinary deck of 52 cards. Let B be
the event that both cards are aces, let S be the event that the ace of spades is chosen, and let A be the
event that at least one ace is chosen. Find (a) P(S|A) and (b) P(B|A)