Question

A box is filled with 15 different socks: 10 green and 5 yellow. Three socks are picked from the box (leaving 12 in the box). Let x be the random variable describing the number of green socks that are selected. Let events A and B be described as follows:

A = the event that not all three socks are green.

B = the event where the first sock selected is green.

Find P(A), P(B), and P(A & B). Show any appropriate nCr expressions.

Answer 1

TOTAL = 15

GREEN = 10 ; P(GREEN) = 10/15

YELLOW = 5 ; P(YELLOW)= 5/15

LET X BE THE RANDOM VARIABLE DESCRIBING THE NUMBER OF GREEN SOCKS

P(A) = P( NOT ALL THE THREE SOCKS ARE GREEN) = 1- P(ALL THE THREE SOCKS ARE GREEN)

= 1- (10/15)³

= 0.3333

P(B)= P(FIRST SOCK IS GREEN)

= P( first is green) * P( other two are also green) + P( first is green) * P( one is green and one is yellow) +

P( first is green)* P( other two are yellow)

= (10/15) *(10/15)² + (10/15) ²C₁ (10/15) (5/15) + (10/15) *(5/15)²

⁼ 0.8148

P( A and B) = P( NOT ALL SOCKS ARE GREEN AND FIRST SOCK IS GREEN)

This means that the first socks is green and the other two are either both yellow or one yellow and one green

= P( first is green) * P( one is green and one is yellow) +  P( first is green)* P( other two are yellow)

=(10/15) ²C₁ (10/15) (5/15) + (10/15) *(5/15)²

=0.3703