Question

According to a reliable source, 65% of murders are committed with a firearm. Suppose 15 murders are randomly selected. First construct a relative and cumulative frequency distribution for the situation. Then confirm that it is both a probability and binomial probability distribution.

a. Compute the mean
b. Compute the standard deviation
c. Find the probability that exactly 10 murders are committed with a firearm.
d. Find the probability that at most 11 murders are committed with a firearm.
e. Find the probability that at least 12 murders are committed with a firearm
f. Find the probability that between 9 and 13 murders are committed with a firearm.

Answer 1

we have p = 65% or 65/100 = 0.65 and n= 15

(A) Mean = n*p

we have n = 15 and p = 0.65

this gives, mean = 15*0.65 = 9.75

(B) Standard deviation =

setting n = 15 and p= 0.65

we get, standard deviaton =

(C) Using the excel function BINOMDIST(number_s,trials,probability,cumulative)

We have to find the probability of exactly 10 murders are committed with a firearm

so, we have number_s = 10, trials = 15, probability = 0.65 and cumulative = 0(for exact probability)

we get

BINOMDIST(10,15,0.65,0) = 0.2123

(D) Using the excel function BINOMDIST(number_s,trials,probability,cumulative)

We have to find the probability that at most 11 murders are committed with a firearm

so, we have number_s = 11, trials = 15, probability = 0.65 and cumulative = 1(for at most probability)

we get

BINOMDIST(11,15,0.65,1) = 0.8273

(E) Probability of at least 12 murders = 1 - Probability of at most 11 murders = 1 - 0.8273 = 0.1727

(F) Probability that between 9 and 13 murders are committed with a firearm = P(x1<x<x2)

using the formula

setting x1 = 9,x2= 13, = 9.75 and = 1.85

we get

so, the required probability is 0.6199