Answer:
![P(X=x)=C(15,x)\cdot(0.75)^x\cdot(0.25)^{(15-x)}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3DC%2815%2Cx%29%5Ccdot%280.75%29%5Ex%5Ccdot%280.25%29%5E%7B%2815-x%29%7D)
Explanation:
The <em>probability mass function</em> (<em>PMF</em>), or <em>frequency function</em>, is the function that gives the probabilities that a <em>discrete random variable</em> take some values.
In this problem, it is requested the frequency function (PMF) for the <em>number of purchasers, among the next 15, who select a chain-driven model</em>.
Then , you need to find, the function that gives P(X=0), P(X=1), P(X=2), P(X=3), . . . up to P(X=15).
Such as any function, the frequency function can be presented as a formula, as a table, or as a graph.
Note that the statement represents a binomial disbribution in which success is that a customer select a chain-driven model and the fail is that a cusotmer does not select a chain-driven model.
The binomial probability for X = the number among the 15 purchasers who select the chain-driven model is given by the formula:
![P(X=x)=C(n,x)\cdot(p)^x\cdot(1-p)^{(n-x)}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3DC%28n%2Cx%29%5Ccdot%28p%29%5Ex%5Ccdot%281-p%29%5E%7B%28n-x%29%7D)
Where:
- n is the number of times the experiment is performed: 15 in our problem
- p is the probability of succes: 0.75 in our problem
- 1-p is the probability of fail: 0.25 in our problem
Then, substitute:
![P(X=x)=C(15,x)\cdot(0.75)^x\cdot(0.25)^{(15-x)}](https://tex.z-dn.net/?f=P%28X%3Dx%29%3DC%2815%2Cx%29%5Ccdot%280.75%29%5Ex%5Ccdot%280.25%29%5E%7B%2815-x%29%7D)
That is the frequency function.
If you want to give it as a table you must find P(X=1), P(X=2), P(X=3), . . . up to P(X=15) using that function. That is not part of the question.