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E(X)=(5-a)(28/487)+(2-a)(144/487)+(-a)(315/487)=0
Therefore, by regrouping the terms after distribution we have:
5(28/487)+2(144/487) - (28/487+144/487+315/487) a = 0
Note that the coefficient of a is the sum of the probablities, which is one. So we have
a = 5(28/487)+2(144/487)= $ 428/487, which is almost equal to 0.8789 or 88 cents, as mentioned
above.
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More precisely, we could run out of red marbles only after three trials, so there
might not be any reds left for the fourth trial.
An even better answer here would be the fact that selection without replacement
makes even the second trial dependent to the first one.
Remark: The experiment will be binomial if we change "without" to "with."
For the interested student: The experiment in part (d) is called "hypergeometric",
only because the selection is "without replacement." Hypergeometric distributions
are different than binomial distributions. Please see the Wikipedia page on
hypergeometric distribution.
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