A ball is chosen at random from a bag containing 150 balls t

A ball is chosen at random from a bag containing 150 balls that are either red or blue and either dull or shiny. There are 36 red shiny balls and 54 blue balls. What is the probability of the chosen ball being shiny conditional on it being red? What is the probability of the chosen ball being dull conditional on it being red?

Solution

Given the problem statement, we can first compute: P(red & shiny) = 36/150 = 0.24 P(blue) = 54/150 = 0.36 Since there are only red and blue balls, the number of red balls (both shiny and dull) are 150-54 = 96 => P(red) = 96/150 = 0.64 Since a ball is either shiny or dull, and we know that there are 96 red balls and 36 red & shiny balls, there must be 96-36 = 60 red & dull balls, and thus P(red & dull) = 60/150 = 0.4 Recall the probability identity, P(A|B) = P(A&B)/P(B), and now computedull; (i) P(shiny|red) = P(shiny&red)/P(red) = 0.24/0.64 = 0.375 (ii) P(dull|red) = P(dull&red)/P(red) = 0.4/0.64 = 0.625