Question

For an urn containing 4 red balls and 6 green balls, let the number of balls randomly drawn be the number of heads turning up when 5 fair coins have been previously flipped. What is the probability of drawing 3 green balls?

Answer 1

X : Number of heads turning up when 5 fair coins have been previously flipped

Possible values of X = 0,1,2,3,4,5

Y : Number of Green balls randomly drawn from the urn

Probability of drawing 3 green balls = P(Y=3)

When X =0,1,2 i.e When number of heads turning up 0,1,2 i.e. number of balls drawn is 0,1,2: Y : Number of green balls drawn can not be 3;

Therefore,

When X=3,4,5 i.e When number of heads turning up 3,4,5 i.e. number of balls drawn is 3,4,5 : Y : Number of green balls drawn can be 3;: Y can be 3;

i.e

Y = 3 ; equivalent :

Y=3 ; when number of heads turning up 3 i.e. Y=3|X=3 or

Y=3 ; when number of heads turning up 4 i.e. Y=3|X=4 or

Y=3 ; when number of heads turning up 5  i.e. Y=3|X=5 or

Therefore:

P(Y=3) = P(X=3)P(Y=3|X=3) + P(X=4)P(Y=3|X=4) +P(X=5)P(Y=3|X=5)

X : Number of heads turning up when 5 fair coins have been previously flipped

p: Probability of a head turning up when a fair coin is flipped = 1/2 = 0.5

Number of fair coins flipped = 5

X follows binomial distribution with n=5 ; p=0.5 (q=1-p=1-0.5=0.5)

Number of red balls in the urn = 4

Number of green balls in the urn = 6

Total number of balls in the urn = 4+6=10

P(Y=3|X=3)

= Probability of getting 3 green balls when 3 balls are drawn from the urn

= Number of ways of drawing 3 green balls from 6 green balls / Number of ways of drawing 3 balls from 10 balls

P(Y=3|X=3) = 0.1667

P(Y=3|X=4)

= Probability of getting 3 green balls when 4 balls are drawn from the urn

= Number of ways of drawing 3 green balls from 6 green balls and 1 red ball from 4 red balls / Number of ways of drawing from 4 balls from 10 balls

P(Y=3|X=4) = 0.3810

P(Y=3|X=5)

= Probability of getting 3 green balls when 5 balls are drawn from the urn

= Number of ways of drawing 3 green balls from 6 green balls and 2 red ball from 4 red balls / Number of ways of drawing from 5 balls from 10 balls

P(Y=3|X=5) = 0.4762

P(Y=3) = P(X=3)P(Y=3|X=3) + P(X=4)P(Y=3|X=4) +P(X=5)P(Y=3|X=5)

= 0.3125x0.1667 + 0.15625x0.3810 + 0.03125x0.4762=0.1265

Probability of drawing 3 green balls = 0.1265