Question

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 25 incoming calls. (Round your answers to three decimal places.) (a) What is the probability that at most 7 of the calls involve a fax message

Answer 1

Answer: 0.726

Explanation:

This will be solved by the probability distribution formula for random variables where combination formula for selection is used.

When determining the probability of random variable X, and when X=r,then

P(X=r) = nCr × p^r × q^n-r

Where n = number of sample And in this case, n = 25

r = desired outcome of sample

p = probability of having a successful outcome. And In this Case, p=25%= 0.25

q = 1-p=1-0.25=0.75

To get probability of at most 7 of the samples being fax, we vary the value of x from 0 to 7 And sum up the different values of probability gotten.

When x = 0

P(X=0) = 25C0 × 0.25^0 × 0.75^25 =0.00075

When x=1

P(X=1) = 25C1 × 0.25¹ × 0.75^24 = 0.00627

When x=2

P(X=2) = 25C2 × 0.25² × 0.75^23 = 0.0251

When x=3

P(X=3) = 25C3 × 0.25³ × 0.75^22 = 0.0641

When x = 4

P(X=4) = 25C4 × 0.25^4 × 0.75^21 =0.1175

When x=5

P(X=5) = 25C5 × 0.25^5 × 0.75^20 = 0.1645

When x=6

P(X=6) = 25C6 × 0.25^6 × 0.75^19 = 0.1828

When x=7

P(X=7) = 25C7 × 0.25^7 × 0.75^18 = 0.1654

Hence, P(X≤7) = 0.00075 + 0.00627 + 0.0251 + 0.0641 + 0.1175 + 0.1645 + 0.1828 + 0.1654 = 0.726(3dp)