A department store sells sport shirts in three sizes small m

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 combinations. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt? What is the probability that the next shirt sold is a medium print shirt? What is the probability that the next shirt sold is a short-sleeved shirt? A long-sleeved shirt? What is the probability that the size of the next shirt sold is medium? What is the probability that the pattern of the next shirt sold is a print? Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium? (Round your answer to three decimal places.) Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Long-sleeved? (Round your answer to three decimal places.) short-sleeved long-sleeved

Solution

(a)

P(long-sleeved and M and Pr) = 0.07

(b)

Since either shirt is long sleeved or short sleeved, cannot be both so the probability that next shirt sold is a medium print shirt will be

P(medium print shirt) = P(long-sleeved and M and Pr) + P(short-sleeved and M and Pr) = 0.07+0.05 = 0.12

(c)

The probability that next short sold is short sleeved will be sum of all probabilites of Short-Sleeved table. So

P(short-sleeved) = 0.04+0.02+0.05+0.09+0.05+0.12+0.03+0.07+0.08 = 0.55

Likewise

P(long-sleeved) = 0.45

(d)

P(M) = P(Medium) = 0.09+0.07+0.07+0.09+0.05+0.12 = 0.49

P(Pr) = P(Print) = 0.02+0.05+0.07+0.02+0.07+0.02 = 0.25

(e)

P( short-sleeved and medium) = 0.09+0.05+0.12 = 0.26

P(medium) = 0.49

So the required probability is

P(medium | short-sleeved) = P(short-sleeved and medium) / P(medium) = 0.26 / 0.49 = 0.5306

(f)

P(short-sleeved) = 0.55

P(long-sleeved) = 0.45

P( short-sleeved and medium) = 0.09+0.05+0.12 = 0.26

P( long-sleeved and medium) = 0.09+0.07+0.07 = 0.23

P( short-sleeved | medium ) = P(short-sleeved and medium) / P( short-sleeved) = 0.26 / 0.55 = 0.4727

P( long-sleeved | medium ) = P(long-sleeved and medium) / P( long-sleeved) = 0.23 / 0.45 = 0.5111