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: 0.726

Step-by-step 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)