Section 1.4 The conditional probability of an event A given that event B has occurred (or will occur) is defined by P(A  B) P(A | B) = ————provided that P(B) > 0. P(B) The probability that two events A and B both occur is defined by the multiplication rule which states that P(A  B) = P(A) P(B | A) = P(B) P(A | B). 1. An urn contains seven red chips labeled distinctively with the integers 1 through 7, eight blue chips labeled distinctively with the integers 1 through 8, nine white chips labeled distinctively with the integers 1 through 9.

(a) P(A) = P(B) =P(C) = P(A  B) = P(B  C) =P(A  C) = P(A  B) = P(B  C) =P(A  C) = P(A | B) = P(B | A) = P(B | C) = P(C | B) = 13/248/24 = 1/33/24 = 1/8 4/24 = 1/61/240 17/2410/24 = 5/12 16/24 = 2/3 4/8 = 1/24/13 1/31/8 One chip is randomly selected. The following events are defined: A = the selected chip is labeled with an odd integer, B = the selected chip is blue, C = the selected chip is labeled with a "4". Find the following probabilities:

(b) P(A) =P(B | A) =P(C | A) = P(A  B) =P(A  C) =P(B) = P(A | C) =P(C) = Two chips are randomly selected without replacement. The following events are defined: A = the first chip is red, B = the second chip is red, C = the second chip is white. Find the following probabilities: 7/246/239/23 (7)(6) ——— (24)(23) (7)(9) ——— (24)(23) 7/24 7/239/24 = 3/8

(c) P(A) =P(B | A) =P(C | A) = P(A  B) =P(A  C) =P(B) = P(A | C) =P(C) = 7/24 9/24 = 3/8 7 2 —– 24 2 (7)(9) ——– /24 9/24 = 3/8 Two chips are randomly selected with replacement. The following events are defined: A = the first chip is red, B = the second chip is red, C = the second chip is white. Find the following probabilities:

(d)Successive chips are randomly selected without replacement. Find the probability that the fourth white chip appears on the seventh draw. A = exactly three white chips are selected in the first six draws B = a white chip is selected on the seventh draw P(A  B) = P(A) P(B | A) = (Note that one could also find P(B) P(A | B).)

(e)Successive chips are randomly selected with replacement. Find the probability that the fourth white chip appears on the seventh draw. A = exactly three white chips are selected in the first six draws B = a white chip is selected on the seventh draw P(A  B) = P(A) P(B | A) = ! —— 3! 24 9 (Note that one could also find P(B) P(A | B).)

2. (a) An urn contains three red marbles and seven white marbles. A drawer contains two red marbles and three white marbles. One marble is randomly selected and transferred from the urn to the drawer. One marble is then randomly selected from the drawer. Find each of the following: the probability that the selected marble is red. A = the marble selected from the drawer is red B = the marble transferred from the urn to the drawer is red B´ = the marble transferred from the urn to the drawer is white P(A) = P((A  B)  (A  B´)) =P(A  B) + P(A  B´) = P(B) P(A | B) + P(B´) P(A | B´) = 3 — 10 1 — — 10 1 — = 3 23 — 60

(b) the probability that the marble transferred from the urn to the drawer is white given that the selected marble is red. P(B´ | A) = P(B´  A) ———— = P(A) (7/30) ———— = (23/60) 14 — 23