Four jelly beans are selected, one at a time from a bowl containing 6 black, 6 red and 6 green jelly beans. Let x represent the number of black jelly beans selected in 4 draws from the bowl.
(A) If this experiment is completed without replacing the jelly beans, explain why x is not a binomial random variable.
(B) If this experiment is completed with replacement of the jelly beans, explain why x is a binomial random variable.
(A) In this experiment, the jelly beans are not replaced after each draw. This means that the probability of drawing a black jelly bean changes for each subsequent draw. As a result, the number of black jelly beans selected in the four draws is not independent of each other. In a binomial random variable, each trial must have the same probability of success and the trials must be independent. Since these conditions are not met in this experiment, x is not a binomial random variable.
(B) If the experiment is completed with replacement, it means that after each draw, the jelly bean is put back into the bowl before the next draw. In this case, the probability of drawing a black jelly bean remains the same in each draw, as the number of jelly beans of each color remains constant throughout the experiment. Additionally, each draw is independent of the others, as the outcome of one draw does not affect the next draw. Therefore, in this scenario, x represents the number of black jelly beans selected in four independent trials with a constant probability of success. Hence, x is a binomial random variable.