Question

A box contains one yellow, two red, and three green balls. Two balls are randomly chosen without replacement. Define the following events:

A: \{ One of the balls is yellow \}

B: \{ At least one ball is red \}

C: \{ Both balls are green \}

D: \{ Both balls are of the same color \}

Find the following conditional probabilities: (a) P(B|D^c)

(b) P(D|C)

(c) P(A|B)

Answer 1

variables :

Y = no. of yellow balls

R = no. of red balls

G = no. of green balls

a.

events :

A : Y=1

B : R >= 1

C :G = 2

D : R=2 or G=2

{Y=2 not included as there is only 1 yellow ball}

b.

P(D|C) if C is given it means both ball are given then D ,i.e. , both balls same color will always be fulfilled

therefore,

P(D|C) = 1

c.

P(A and B) = P(1st ball red)*P(2nd ball yellow) + P(1st ball yellow)*P(2nd ball red)

= (2/6)*(1/5) + (1/6)*(2/5)

= 0.1333

P(B) = P(1 red) + P(2 red)

= (select one red ball out of 2red balls)*(select one non red ball out of 4 non red balls)/(select 2 balls from 6) + (select 2 red balls from 2 red balls)/(select 2 balls from 6)

= (2C1)*(4C1)/(6C2) + (2C2)/(6C2)

= 2*4/15 + 1/15

= 9/15

= 3/5

P(B) = 0.6

P(A|B) = P(A and B)/P(B)

= 0.1333/0.6

P(A|B) = 0.2222

(please UPVOTE)