Question

120 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. What is the probability that 1 a. At most 6 of the calls involve a fax message? 2 b. Exactly 6 of the calls involve a fax message? 3 c. At least 6 of the calls involve a fax message? 4 d. More than 6 of the calls involve a fax message? 5 e. The number of calls among the 25 that involve a fax transmission

Answer 1

Answer:

a)0.5611

b)0.1828

c)0.6217

d)0.4389

e)6.25

Step-by-step explanation:

Let's define the following event :

X : ''Calls involving a fax message''

The random variable X can be modeled as a Binomial random variable.

N ~ Bi(n,p)

N ~ Bi(25,0.25)

Where n is the sample and p is the success probability

Let's also define nCr as the combinatorial number :

$nCr=\frac{n!}{r!(n-r)!}$

The probability function for X is :

$P(X=k)=nCk.p^{k}.(1-p)^{n-k}$

Where k is the number of success

a) $P(X\leq 6)$

$P(X\leq 6)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)$

$P(X\leq 6)=0.5611$

b)$P(X=6)=25C6.(0.25)^{6}.(1-0.25)^{19}=0.1828$

c) $P(X\geq 6)=1-P(X$

d)$P(X>6)=1-P(X\leq 6)=1-0.5611=0.4389$

e) They are asking us about the expected value for random variable X

In a Binomial random variable :

E(X) = μ = n.p

$E(X)=25(0.25)=6.25$