Question

Ten percent of the corporate managers at Axolotl Industries majored in humanities. What is the expected number of managers to be interviewed before finding the first one with a humanities major?

• 15

• 20

• 10

• 17

Answer 1

Solution

Back-up Theory

If a discrete random variable, X, has probability function, p(x), x = x₁, x₂, …., x_n, then

Expected value of X = E(X) = Σ{x.p(x)} summed over all possible values of x….................................................…. (1)

Now, to work out the solution,

Let X = number of managers to be interviewed before finding the first one with a humanities major

Ten percent of the corporate managers at Axolotl Industries majored in humanities =>

Probability of finding a manager with a humanities major = 0.1 and hence

Probability of not finding a manager with a humanities major = 0.9

Now, X = 1 => first manager interviewed is a humanities major and so P(X = 1) = 0.1

X = 2 => first manager interviewed is not a humanities major and the second manager interviewed

is a humanities major. So P(X = 2) = 0.9 x 0.1

X = 3 => first two manager interviewed are not humanities majors and the third manager interviewed

is a humanities major. So P(X = 3) = 0.9² x 0.1

Similarly, P(X = 4) = 0.9³ x 0.1; P(X = 5) = 0.9⁴ x 0.1; and so on, the process, mathematically, would

go to infinity.

So, vide (1),

E(X), say =

E = (1 x 0.1) + (2 x 0.9 x 0.1) + (3 x 0.9² x 0.1) + (4 x 0.9³ x 0.1) + ................... to infinity ............................ (2)

0.9E =              (1 x 0.9 x 0.1) + (2 x 0.9² x 0.1) + (3 x 0.9³ x 0.1) + ................... to infinity ............................ (3)

(2) - (3):

0.1E = (1 x 0.1) + (1 x 0.9 x 0.1) + (2 x 0.9² x 0.1) + (3 x 0.9³ x 0.1) + ................... to infinity

= (1 x 0.1)/(1 – 0.9) [applying the formula for the sum of an infinite geometric series]

= 1

=> E = 1/0.1 = 10. Thus,

expected number of managers to be interviewed before finding the first one with a humanities major = 10 Answer

Third Option

DONE

[going beyond,

X has Geometric Distribution with parameter p = 0.1 and mean of Geometric Distribution is 1/p.]