Question

In: Statistics and Probability

A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid,...

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.

Short-sleeved
Pattern
Size Pl Pr St
S 0.04 0.02 0.05
M 0.09 0.05 0.12
L 0.03 0.07 0.08
Long-sleeved
Pattern
Size Pl Pr St
S 0.03 0.02 0.03
M 0.08 0.08 0.07
L 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? A long-sleeved shirt?

short-sleeved shirt     
long-sleeved shirt



(d) 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?


(e) 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.)


(f) 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

Solutions

Expert Solution

(a) The probability that the next shirt is a medium,

long-sleeved print shirt = 0.08. (Ans).

(b) The probability that the next shirt sold is a medium

print shirt = 0.05 + 0.08 = 0.13. (Ans).

(c) The probability that the next shirt sold is a short sleeved

shirt = 0.04 + 0.02 + 0.05 + 0.09 + 0.05 + 0.12 + 0.03 + 0.07

+ 0.08 = 0.55. (Ans).

The probability that the next shirt sold is a medium

= 1 - 0.55 = 0.45. (Ans).

(d) The probability that the size of the next shirt is medium

= 0.09 + 0.05 + 0.12 + 0.08 + 0.08 + 0.07 = 0.49. (Ans).

The probability that the next shirt is print

= 0.02 + 0.05 + 0.07 + 0.02 + 0.08 + 0.02 = 0.26. (Ans).

e) P(Medium | Short Sleeved)

= P(Medium and Short Sleeved) / P(Short Sleeved)

= (0.09 + 0.05 + 0.12) / 0.55 = 0.26 / 0.55 = 0.4727. (Ans).

f) P(Short sleeved | Medium)

= P(Short sleeved and Medium) / P(Medium)

= 0.26 / 0.49 = 0.5306. (Ans).

P(Long sleeved | Medium)

= P(long sleeved and Medium) / P(Medium)

= (0.08 + 0.08 + 0.07) / 0.45 = 0.23 / 0.45 = 0.5111. (Ans)

orchestra answered 1 year ago
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