A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and
two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinationsShort-sleeved Pattern Size Pl Pr StS 0.04 0.02 0.05M 0.07 0.10 0.12L 0.03 0.07 0.08Long-sleeved Pattern Size Pl Pr StS 0.03 0.02 0.03M 0.06 0.07 0.07L 0.04 0.02 0.08(a) What is the probability that the next shirt sold is a medium, long-sleeved, print shirt?(b) What is the probability that the next shirt sold is a medium print shirt?(c) What is the probability that the next shirt sold is a short-sleeved shirt?What is the probability that the next shirt sold is a long-sleeved shirt?(d) What is the probability that the size of the next shirt sold is a medium?What is the probability that the pattern of the next shirt sold is a print?(e) Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium?(f) Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? What is the probability that it was long-sleeved?
1 answer:
Answer and Step-by-step explanation:
a) probability of selling a medium, long sleeved printed shirt, P(M ∩LS ∩PR) = 0.07, directly from the table of probabilities
b) probability that the next shirt sold is a medium printed shirt, (M ∩Pr) = P(M,Pr,LS) + P(M,Pr,SS) = 0.07+0.10 = 0.17
c) probability that the next shirt is a short sleeved shirt, P(SS) = sum of 9 probabilities in Short Sleeved shirt table = 0.58
probability that the next shirt is a long sleeved shirt, P(LS) = 1 - 0.58 = 0.42 or add up the total probabilities in the long sleeved shirt table, P(LS) = sum of 9 probabilities in the long sleeved shirt table = 0.42
d) probability that the next shirt is a medium, P(M) = P(M,SS) + P(M,LS) = (0.07+0.10+0.12) + (0.06+0.07+0.07) = 0.49
probability that the next shirt sold is a print, P(Pr) = P(Pr,SS) + P(Pr,LS) = (0.02+0.10+0.07) + (0.02+0.07+0.02) = 0.40
e) probability that the shirt sold is a medium given that the shirt just sold was a short-sleeved plaid, P(M|SS,PL) = (P(M,SS,PL))/P(SS,PL) = 0.07/(0.04+0.07+0.03) = 0.5
f) probability that the shirt sold is short sleeved given that the shirt just sold was a medium plaid, P(SS|M,PL) = (P(M,SS,PL))/P(M,PL) = 0.07/(0.07+0.06) = 0.53846 = 0.54
probability that the shirt sold is long sleeved given that the shirt just sold was a medium plaid, P(LS|M,PL) = (P(M,LS,PL))/P(M,PL) = 0.06/(0.07+0.06) = 0.462 = 0.46
QED!
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