Question

You have a jar that contains 4 pink marbles, 3 yellow marbles, and 5 gray marbles.

You will take out a marble at random and then replace it. You will do this a total of

8 times.

(a) Given that you observe at least 6 pink marbles, what is the probability that you

observe 1 yellow marble and 1 gray marble?

Answer 1

The jar contians 4 pink , 3 yellow , 5 gray marbel .

Probability that I observe pink ball = 4/12

Probability that I observe yellow ball = 3/12

Probability that I observe gray ball = 5/12

A= event that I observe atleast 6 pink marvel out of my 8 take

B= Event I observe 1 yellow , 1 gray ball out of my 8 take

I have to find P ( B|A) = P ( B A ) / P (A )

P(B A ) =I observe 6 pink 1 yellow 1 gray out of my 8 take

This follows multinomial distribution

P(B A ) = 8 !* (4/12 )^6 * (3/12 )* (5/12) / ( 6! * 1 ! * 1 !) = .008

P(A) = P(I observe 6 pink 1 yellow 1 gray out of my 8 take ) + P(I observe 6 pink 2 yellow out of my 8 take ) + P(I observe 6 pink 2 gray out of my 8 take )+ P(I observe 7 pink 1 yellow out of my 8 take ) + P (I observe 7 pink 1 gray out of my 8 take ) +P(I observe 8 out of my 8 take )

= 8 !* (4/12 )^6 * (3/12 )* (5/12) / ( 6! * 1 ! * 1 !) + 8 !* (4/12 )^6 * (3/12) ^2 / ( 6! * 2!)

+ 8 !* (4/12 )^6 * (5/12) ^2 /( 6! * 2!) + 8 !* (4/12 )^7 * (3/12) / ( 7! * 1!)+ 8 !* (4/12 )^7 * (5/12) / ( 7! * 1!)+ (4/12 )^8

=.008 + .0024 + .0067 + .0009 + .0015 + .00015= 0.01965

P ( B|A) = .008 / 0.01965 = 0.407 [Ans]

Given that you observe at least 6 pink marbles, the probability that I observe 1 yellow marble and 1 gray marble is 0.407.