Question

Conditional Probability Problem: An urn contains 5 red balls, 4 green balls, and 4 yellow balls for a total of 13 balls. If 5 balls are randomly selected without replacement, what is the probability of selecting at least two red balls given that at least one yellow ball is selected? Please show all steps.

Answer 1

P(at least one yellow ball )=1-P(no yellow ball) 1-P(select 5 balls from remaining 9 balls)=1-⁹C₅/¹³C₅ =1-126/1287=1161/1287

P(at least one yellow and at least 2 red balls)=P(1 yellow , 2 red , 2 green)+P(2 yellow , 2 red and 1 green)+P(3 yellow,2 red)+P(1 yellow , 3 red , 1 green)+P(1 yellow , 4 red)+P(2 yellow , 3 red)

=(⁵C₁*⁴C_2*⁴C₂ +⁵C₁*⁴C_2*⁴C₂ +⁵C₁*⁴C_2*⁴C₂ +⁵C₁*⁴C_2*⁴C₂ +⁵C₁*⁴C_2*⁴C₂ )/¹³C₅ =605/1287

hence P( selecting at least two red balls given that at least one yellow ball is selected )=(605/1287)/(1161/1287)

=605/1161