Answer to Question #326326 in Statistics and Probability for VITHU

Question

A student majoring in marketing is trying to decide on the number of firms to which she

should apply. Given her work experience, grades, and extra curricular activities, she has

been told by a placement counselor that she can expect to receive a job offer from 80% of

the firms to which she applies. Wanting to save time, The student applies to only five firms.

Assuming the counselor’s estimate is correct, find the probability that the student receives

at least three offers.

Answer 1

The probability that the student receives an offer $p =0.8,$ that she doesn't $q = 1-0.8=0.2.$

Obviously the number of offers X may be any value of 0, 1, 2, 3, 4, 5.

We have a Bernoulli trial - exactly two possible outcomes, "success" (the student receive an offer) and "failure" (she doesn't receive) and the probability of success is the same every time the experiment is conducted (the student applies to a firm).

The probability of each result

$P(X=k)=\begin{pmatrix}n\\k\end{pmatrix}\cdot p^k\cdot q^{n-k}=\\ =\begin{pmatrix}5\\k\end{pmatrix}\cdot 0.8^k\cdot 0.2^{n-k}=\\ =\cfrac{5!}{k!\cdot(5-k)!}\cdot 0.8^k\cdot 0.2^{n-k}.$

The sought probability that the student receives at least three offers:

$P(X\geq 3)=\\ =P(X=3)+P(X=4)+P(X=5)=\\ =\cfrac{5!}{3!\cdot2!}\cdot 0.8^{3}\cdot 0.2^2+\cfrac{5!}{4!\cdot1!}\cdot 0.8^{4}\cdot 0.2^1+\\ +\cfrac{5!}{5!\cdot0!}\cdot 0.8^{5}\cdot 0.2^0=\\ =0. 94208.$