Mathematics, 10.03.2020 16:57 JOIful

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 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?(b) What is the probability that exactly 7 of the calls involve a fax message?(c) What is the probability that at least 7 of the calls involve a fax message?(d) What is the probability that more than 7 of the calls involve a fax message?

Answers: 3

Show answers

Answers

Answer from: amelvin41

a) 0.968 = 96.8% probability that at most 7 of the calls involve a fax message

b) 0.055 = 5.5% probability that exactly 7 of the calls involve a fax message

c) 0.913 = 91.3% probability that at least 7 of the calls involve a fax message

d) 0.032 = 3.2% probability that more than 7 of the calls involve a fax message

Step-by-step explanation:

For each call, there are only two possible outcomes. Either there is a fax message, or there is not. The probability of a call being a fax is independent from other calls. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

In this problem we have that:

p = 0.2, n = 20

(a) What is the probability that at most 7 of the calls involve a fax message?

This is $P(X \leq 7)$ .

$P(X \leq 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)$

In which

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{20,0}.(0.2)^{0}.(0.80)^{20} = 0.0115$

$P(X = 1) = C_{20,1}.(0.2)^{1}.(0.80)^{19} = 0.0576$

$P(X = 2) = C_{20,2}.(0.2)^{2}.(0.80)^{18} = 0.1369$

$P(X = 3) = C_{20,3}.(0.2)^{3}.(0.80)^{17} = 0.2054$

$P(X = 4) = C_{20,4}.(0.2)^{4}.(0.80)^{16} = 0.2182$

$P(X = 5) = C_{20,5}.(0.2)^{5}.(0.80)^{15} = 0.1746$

$P(X = 6) = C_{20,6}.(0.2)^{6}.(0.80)^{14} = 0.1091$

$P(X = 7) = C_{20,7}.(0.2)^{7}.(0.80)^{13} = 0.0545$

$P(X \leq 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) = 0.0115 + 0.0576 + 0.1369 + 0.2054 + 0.2182 + 0.1746 + 0.1091 + 0.0545 = 0.968$

0.968 = 96.8% probability that at most 7 of the calls involve a fax message

(b) What is the probability that exactly 7 of the calls involve a fax message?

$P(X = 7) = C_{20,7}.(0.2)^{7}.(0.80)^{13} = 0.0545$

Rounded to three decimal places

0.055 = 5.5% probability that exactly 7 of the calls involve a fax message

Either less than 7 of the calls involve a fax message, or at least 7 does. The sum of the probabilities of these events is 1. So

$P(X < 7) + P(X \geq 7) = 1$

We want $P(X \geq 7)$ . So

$P(X \geq 7) = 1 - P(X < 7)$

In which

P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) = 0.0115 + 0.0576 + 0.1369 + 0.2054 + 0.2182 + 0.1746 + 0.1091 = 0.913

0.913 = 91.3% probability that at least 7 of the calls involve a fax message

(d) What is the probability that more than 7 of the calls involve a fax message?

Either 7 or less calls involve a fax message, or at least 7 do. The sum of the probabilities of these events is decimal 1. So

$P(X \leq 7) + P(X 7) = 1$

We want P(X > 7).

From a, we have that $P(X \leq 7) = 0.968$ . So

$P(X \leq 7) + P(X 7) = 1$

0.968 + P(X 7) = 1

P(X 7) = 0.032

0.032 = 3.2% probability that more than 7 of the calls involve a fax message

Answer from: Quest

it's both, now can you answer my most recent question?

Answer from: Quest

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 18:30

The clayton family’s pool has vertices at the coordinates (0, 2), (0, 5), (2, 5), (2, 6), (5, 6), (5, 1), (2, 1), and (2, 2). if each grid square has an area of 9 square feet, what is the area of the pool?

Answers: 1

Answer

Mathematics, 21.06.2019 22:00

Aschool had an election where the candidates received votes in the ratio 1: 2: 3. if the winning candidate received 210 votes, how many total people voted in the election?

Answers: 1

Answer

Mathematics, 21.06.2019 22:30

The ivring berlin learning center would like a possible net profit of $650,000 on its new resource manual it sells to schools. the fixed costs for producing the manual are $780,000 and the variable cost per package is $19.85. if the estimated unit sales are 1,250,000 units, what selling price unit should the ivring berlin learning center try?

Answers: 1

Answer

Mathematics, 21.06.2019 23:30

The longer leg of a 30-60-90 trangle is 18

Answers: 1

Answer

You know the right answer?

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 20%...

Questions