When I covered the binomial formula, I showed you how the formula could be gotten from a pattern in the probability tree.

For this last extra credit assignment, I want you to derive the formula for a special type of 'negative binomial'.

Here n (the number of trials, or levels of branches) is NOT fixed. However we are still in a situation where we have two branches: one is 'success' with probability p, and the other the 'failure' with probability (1-p).

The problem is this: What is the probability that first boy child occurs at the xth child in a family.

In other words, what is the probability that the first boy child is the first child in a family?

What is the probability that the first boy child is the second child in a family?

What is the probability that the first boy child is the third child in a family?

(I'm choosing boys because both sections chose girls for the binomial 'success' in class.)

The probability of a boy is 0.51.

a.Draw a tree with 10 levels of branches, following the pattern below, on a separate page (portrait only, please):

Draw the probability tree as follows (put the boy branch on top and girl on bottom at every level, you'll soon see why):

i)First set of branches, boy and girl. What is the chance of a boy being the first child?

ii)Second set of branches - CONTINUE ONLY WITH THE GIRL BRANCH - WE DO NOT HAVE TO CONTINUE THE BOY BRANCH ANY MORE - WE GOT THE FIRST BOY ON THAT ONE.

Second set of branches, boy branch and girl branch.What is the chance of the firstboy being the second child?

iii)Third set of branches: continue only with the girl branch from ii) - we do not have to continue with the previous boy branches any more.

Third set of branches, boy branch and girl branch. What is the chance of the first boy being the third child?

Continue along in the same vein until you have 10 levels of branches.

Fill in the probabilities you got in the table below. Do not calculate them out. Just leave them in terms of p's and (1-p)'s, with p = .51 and 1p = 0.49.

X

P(X

Answer the following questions.

What is the first possible value for X? Why?

From a mathematical point of view: does X have an ending, or can X go on forever? Why?

From a practical point of view, does X have an ending, or can X go on forever? Why?

Does the discrepancy between the mathematical model and the practical point of view mean that the math model is completely useless? Explain.

Can you state what the general formula in terms of P's and (1-P)'s rather than 0.51 and 0.49 should be for situations like this (in other words, the first success occurs on the xth trial?)