Question

Show your steps! The previous answer posted is wrong...could you do it again)

4. Three Plinko chips fall in such a way that each chip is equally likely to fall into any one of five slots. All chips may fall into the same slot or two may fall into the same slot or all three may fall into different slots. Let ? denote the number of unoccupied slots at the end of the game. What is the moment generating function of ??

Answer 1

The number of occupied slots can be either or .

Subtracting each of these numbers from slots, we have, can either be or .

Therefore,

We only need to calculate and .

Now, means three slots are occupied. So, each chip falls into a different slot.

So, is the same as the probability that these chips fall into different slots.

So, we have to choose three slots randomly. This can be done in ways.

Probability that these chips fall into the slots

= (Probability that a chip falls into ) (Probability that a chip falls into ) (Probability that a chip falls into )

=

So,

Similarly, we obtain that and

Simplifying them, we get,

So, moment generating function of is given by