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.
If x is a binomial random variable, then p, the probability of selecting 1
black jelly bean in 1 trial, must not be different in any of the 4 selections
of one jelly bean. If you replace the jelly bean, as in (B), this probability
p will not change, and x will be a binomial random variable. However if you do
not replace the jelly bean each time, as in (A), the probability will change each
time you draw a jelly bean, and thus (A) will not allow x to be a binomial
variable.
Another way of stating it is that if you do not replace the jelly bean each
time, each successive trial (selection) will be DEPENDENT on what you drew
before. However if x is to be a binomial random variable, each selection
(trial) must be INDEPENDENT of any other selection (trial).
Edwin