Negative probability function

Question

You can get a group rate on tickets to a play if you can find 25 people to go. Assume each person you ask responds independently and has a 20% chance of agreeing to buy a ticket. Let X be the total number of people you have to ask in order to find 25 who agree to buy a ticket. Find the probability function of X

This is a negative probability function.

We know $p = 0.2$

The success is $S = \{ \text{Person agrees to buy ticket}\}$ and the failure $F = \{\text{Person doesnt buy ticket} \}$, let

$Y = \{\text{# people that don't buy ticket} \}$ then:

$$Y \sim NB(25, 0.2)$$

Hence

$P(Y = y) = \binom{y + 24}{y}(0.2)^{25}(0.8)^y$

We know total is $X = Y + 25$, thus

$P(X - 25 = y) \implies P(X = y + 25) = \binom{y + 49}{y + 25}(0.2)^{25}(0.8)^{y+25}$

Is this right?

Answer 1

I think you mean Negative Binomial (NB) distribution. Let Y be the number of "No's" you get until you have 25 "Yes", i.e. 25 buyers. Y is then NB-distributed, i.e. $$ P\left( Y=k \right) = \binom{k + r -1}{k} \cdot p_{No}^k \cdot \left( 1 - p_{No} \right)^r $$ Note that $p_{No}$ is the probability to get a "No" which is 0.8, and $r$ is the number of Yes-answers already known to be 25. We can insert $r=25$ and $p_{No}=0.8$ to get: $$ P\left( Y=k \right) = \binom{k + 24}{k} \cdot 0.8^k \cdot 0.2^{25} $$ Here, k runs from 0 to $\infty$ (integers).

Let the total number of questions you have to ask be $X$. Then, $X = Y+25$ because you stop when you have 25 buyers. We have: $$ P\left( X=n \right) = P\left( Y+25=n \right) = P\left( Y=n-25 \right)$$ It follows that $$ P\left( X=n \right) = \binom{n-1}{n-25} \cdot 0.8^{n-25} \cdot 0.2^{25} $$ (replace $k$ by $n-25$ in the 2nd formula)

In the last formula, $n$ runs from 25 to $\infty$. The X distribution is shifted to the right when plotted together with the Y distribution.