Question

A roommate matching service surveys potential tenants for an apartment complex. They run the survey data through a software program that uses an algorithm to identify whether 2 people are a match. The program follows a binomial distribution and finds a positive match 30% of the time. It is able to compare 100 sets of potential roommates per day

a. What is the probability that the software will find exactly 40 matches in any given day?

b. What is the probability that there will be between 24 and 32 matches in any given day?

c. What is the probability that there will be at least 35 matches

d. Is it appropriate to use the normal distribution to approximate probabilities from the binomial distribution of the number of potential roommate matches? How do you know?

e. Use the binomial distribution to find the probability that there will be at most 28 matches found.

f. Approximate the probability from Part (e) above using the normal distribution, if it is appropriate to do so.

Answer 1

Here random variable X is number of matches in any given day.

n is the fixed number of trials and x is the specified number of successes. n – x is the number of failures p is the probability of success on any given trial 1 – p is the probability of failure on any given trial

x ~ Binomial with p = 0.30 and n =100

        x = 0,1, 2,...............,100

where

a) P(X = 40) =

                    =

                   = (100! /40! *60!) * 0.30^40 * 0.70^60

                  = 0.00849     

                  = 0.0085           (Round to 4 decimal places)

Probability that software will find exactly 40 matches in any given day is 0.0085

b)

we can find these probability form excel using command "=BINOM.DIST(x,n = 100 , p =0.30 , cumulative =TRUE)"          

= BINOM.DIST(31,100,0.30,TRUE) = 0.6331         (Round to 4 decimal places)

= BINOM.DIST(23,100,0.30,TRUE) = 0.0755       (Round to 4 decimal places)

         (Round to 4 decimal places)

Probability that there will be between 24 and 32 matches in any given day is 0.5576

c)

Using command in excel "=BINOM.DIST(34,100,0.30,TRUE)" = 0.8371    (Round to 4 decimal places)

           (Round to 4 decimal places)

Probability that there will be at least 35 matches is 0.1629

d) Conditions for Binomial to normal :

n *P and n*(1-p) should be greater than or equal to 5

n * p = 100 *0.30 = 30

n *(1-p) = 100* 0.70 = 70

Here n*p and n*(1-p) are greater than 5 .

Hence we can use normal approximation.

e) Using command "BINOM.DIST(28,100,0.30,TRUE)"

      (Round to 4 decimal places)

Probability that there will be at most 28 matches found is 0.3768

f) Here we can use normal approximation since we proved it above in part (d)

Here we have to use continuity correction factor since Binomial is discrete and Normal is continuous.

         (we have added correction factor 0.5 in 28)

First we have to find Z score

z = -0.3273 = -0.33 (Round to 2 decimal places)

        (Round to 4 decimal places)        

Probability that there will be at most 28 matches found is 0.3707