Question

1. A box contains 12 chocolates, 3 of which are white chocolate, 4 milk chocolate, and 5 dark chocolate.

I am sharing the box with three of my friends. The four of us take turns, each one drawing a chocolate at random from those available in the box. Yes, we're drawing without replacement. We intend to eat the chocolates, not put them back.

Politely, I let my friends draw before me.

For each of the subparts, provide your answer as an unsimplified fractional expression.

a) What is the chance that I draw a dark chocolate?

b) Given that none of my friends draws a dark chocolate, what is the chance that I draw a dark chocolate?

c) What is the chance that all four of us draw milk chocolates?

d) What is the chance that the friend who gets to select first draws a dark chocolate and I draw a dark chocolate too?

Answer 1

(a)

There can be 4 cases: All of the three friends draws dark chocolate with probability 5P3/12P3 = 1/22

Exactly two of the three friends draws dark chocolate with probability

3P1*5P2*7P1/12P3 = 7/22

Exactly one of the three friends draws dark chocolate with probability

3P1*5P1*7P2/12P3 = 21/44

None of the three friends draws dark chocolate with probability

7P3/12P3 = 7/44

The chance that I draw a dark chocolate =

1/22*2/9 + 7/22*3/9 + 21/44*4/9 + 7/44*5/9 = 5/12

(b)

The required probability = 5/9

(c)

The required probability = (4*3*2*1)/(12*11*10*9) = 1/495

(d)

The required probability

= (5*7*6*4 + 5*2*4*7*3 + 5*4*3*2)/(12*11*10*9)

= 5/33