Finding Probabilities of Cell Generations

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Homework Statement

20. The probabilities that a certain type of cell produces during its lifetime 0,1,2 cells of the first
generation are 1/4 , 1/2, 1/4, and any cell of the first generation produces cells of the second
generation under the same conditions. Find the probabilities
(i) that three cells will produce exactly three cells of the first generation?
(ii) that two cells will produce at least one cell of the first generation?
(iii) that one cell will give rise to at least one cell of the second generation?

Homework Equations

The Attempt at a Solution

In fact, I am not quite understanding what's going on...

 

[LCKurtz]

I'll get you started. Let X, Y, and Z be the number of cells produced by each of your original three cells in the first generation. For each of these you know the discrete probability function values:

P(X = 0) = 1/4
P(X = 1) = 1/2
P(X = 2) = 1/4

and similarly for Y and Z. For (a) you are asking for

P(X + Y + Z = 3)

Now X + Y + Z = 3 can happen in several ways. One way would be X = 2, Y = 1, Z = 0. What is the probability associated with that? List all the other ways and calculate the probabilities for them and add them up.