Question

Pierre works 5 days a week. He has 12 shirts, 8 pants, 8 ties, and 4 jackets that he can wear to work. Of these, 4 shirts, 3 pants, 2 ties, and 2 jackets are blue. Each day he randomly selects one of each item to wear. Assume the selections are independent, and assume his butler launders his clothes every night so he has a full closet each morning.

a.) what is the probability Pierre's entire outfit next Monday will be blue?

b.) what is the probability Pierre will wear at least 1 entirely blue outfit during his next 5-day week?

c.) what is the probability that Pierre will wear entirely blue outfits on Monday and Friday while wearing outfits which are not entirely blue on Tuesday through Thursday?

d.) what is the probability Pierre will wear an entirely blue outfit on exactly 2 of the 5 days next week?

Answer 1

a)

probability Pierre's entire outfit next Monday will be blue =P(blue shirt)*P(blue pant)*P(blue tie)*P(blue jacket) =(4/12)*(3/8)*(2/8)*(2/4)=1/64

b)

probability Pierre will wear at least 1 entirely blue outfit during his next 5-day week

=1-P(none of 5 days wear entirely blue outfit)=1-(1-1/64)⁵ =0.075721

c) probability that Pierre will wear entirely blue outfits on Monday and Friday while wearing outfits which are not entirely blue on Tuesday through Thursday =(1/64)*(63/64)*(63/64)*(63/64)*(1/64)

=0.000233

d)

probability Pierre will wear an entirely blue outfit on exactly 2 of the 5 days next week

=⁵C₂(1/64)²(63/64)³ =0.002329