1. A multiple-choice test has 6 questions. There are 4 choices for each question. A student who has not studied for the test decides to answer all questions randomly with a Probability of success = p =.25. Let X represent the number of correct answers out of six questions.
(i) Use binomial distribution to complete the table
(ii) Find the probability that the student will be successful in at least 4 questions.
(iii) Find the probability that the student will be successful in at most 3 problems
(iv) Compute the mean μ and the standard deviation σ for the above problem.
2. A bank recorded the number of ATM transactions made by its customers in one day. The number x of daily ATM transactions per customer can be approximated by the following probability distribution.
|
x
|
P(x)
|
|
0
|
.22
|
|
1
|
.28
|
|
2
|
.20
|
|
3
|
.12
|
|
4
|
.09
|
|
5
|
.07
|
|
6
|
.02
|
Find (i) P(X≤4)
(v) P(2 < X ≤ 5)
(vi) Find the mean of X
(vii) Find the standard deviation of X.
3. A market research team compiled the following discrete probability distribution for families residing in Randolph County. In this distribution x represents the number of evenings the family dines outside their home during a week.
|
x
|
P(x)
|
|
0
|
0.30
|
|
1
|
0.50
|
|
2
|
0.10
|
|
3
|
0.10
|
(i) Calculate the mean (Expected) number of dines outside their home.
(ii) Calculate the variance of the number of dines
(iii) Calculate the standard deviation of the number of dines
4. It is known that the number of business failure in USA per hour is distributed as Poisson with mean µ= 8 .
(i) Find the probability that exactly four businesses will fail in any given hour.
(ii) Find the probability that at least four businesses will fail in any given hour
(iii) Find the probability that more than four businesses will fail in any given hour.
5. On Saturdays, cars arrive at Sami Schmitt's Scrub and Shine Car Wash at the rate of 8 cars per twenty minute interval. Using the Poisson distribution,
(i) find the probability that five cars will arrive during the next five minute interval
(ii) Find the probability that at least three cars will arrive during the next 5 minute interval.
(iii) Find the probability that at no car will arrive during the next 5 minute interval.
6. A volunteer ambulance service handles zero to three service calls on any given day. The probability distribution for the number of service calls is as follows:
X: the number of service calls Probability
0 .15
1 .25
2 .40
3 .20
(i) What is the mean number of service calls per day
(ii) What is the variance of the number of service calls?
(iii) What is the standard deviation of the number of service calls?
(iv) Compute the coefficient of variation of the number of service calls.