Probability - Binomial distribution

1. A random sample of 15 people is taken from a population in which 40% favour a particular political stand. What is the probability that exactly 6 individuals in the sample favour this political stand?
   1. 0.4000
   2. 0.5000
   3. 0.4000
   4. 0.2066
   5. 0.0041
   Solution d
2. Experience has shown that a certain lie detector will show a positive reading (indicates a lie) 10% of the time when a person is telling the truth and 95% of the time when a person is lying. Suppose that a random sample of 5 suspects is subjected to a lie detector test regarding a recent one-person crime. Then the probability of observing no positive reading if all suspects plead innocent and are telling the truth is
   1. 0.409
   2. 0.735
   3. 0.00001
   4. 0.591
   5. 0.99999
   Solution d
3. It has been estimated that about 30% of frozen chicken contain enough salmonella bacteria to cause illness if improperly cooked. A consumer purchases 12 frozen chickens. What is the probability that the consumer will have more than 6 contaminated chickens?
   1. .961
   2. .118
   3. .882
   4. .039
   5. .079
   Solution d Past performance - 89 Dec - 74% Past performance - 90 Oct - 68% Past performance - 92 Oct - 93% Past performance - 97 Aug - 91%
4. Refer to the previous question. Suppose that a supermarket buys 1000 frozen chickens from a supplier. Find an approximate 95% interval for the number of frozen chickens that may be contaminated.
   1. (90, 510)
   2. (285, 315)
   3. (0, 730)
   4. (270, 330)
   5. (255, 345)
   Solution d Past performance - 90 Oct - 74% Past performance - 97 Aug - 81% (13%-b)
5. Which of the following is NOT an assumption of the Binomial distribution?
   1. All trials must be identical.
   2. All trials must be independent.
   3. Each trial must be classified as a success or a failure.
   4. The number of successes in the trials is counted.
   5. The probability of success is equal to .5 in all trials.
   Solution e Past performance - 90 Oct - 84% Past performance - 96 Nov - 97%
6. It has been estimated that as many as 70% of the fish caught in certain areas of the Great Lakes have liver cancer due to the pollutants present. Find an approximate 95% range for the number of fish with liver cancer present in a sample of 130 fish.
   1. (80, 102)
   2. (86, 97)
   3. (63, 119)
   4. (36, 146)
   5. (75, 107)
   Solution a Past performance - 89 Dec - 83% Past performance - 91 Oct - 56% (11%d, 20% e) Past performance - 92 Oct - 78%
7. In a triangle test a tester is presented with three food samples, two of which are alike, and is asked to pick out the odd one by testing. If a tester has no well developed sense and can pick the odd one only, by chance, what is the probability that in five trials he will make four or more correct decisions?
   1. 11/243
   2. 1/243
   3. 10/243
   4. 233/243
   5. 232/243
   Solution a
8. The probability that a certain machine will produce a defective item is 1/4. If a random sample of 6 items is taken from the output of this machine, what is the probability that there will be 5 or more defectives in the sample?
   1. 1/4096
   2. 3/4096
   3. 4/4096
   4. 18/4096
   5. 19/4096
   Solution e
9. The probability that a certain machine will produce a defective item is 0.20. If a random sample of 6 items is taken from the output of this machine, what is the probability that there will be 5 or more defectives in the sample?
   1. .0001
   2. .0154
   3. .0015
   4. .2458
   5. .0016
   Solution e
10. Suppose 60% of a herd of cattle is infected with a particular disease. Let Y = the number of non-diseased cattle in a sample of size 5. The distribution of Y is
    1. binomial with n = 5 and p = 0.6
    2. binomial with n = 5 and p = 0.4
    3. binomial with n = 5 and p = 0.5
    4. the same as the distribution of X, the number of infected
    5. Poisson with = .6
    Solution b
11. Fifteen percent of new residential central air conditioning units installed by a supplier need additional adjustments requiring a service call. Assume that a recent sample of seven such units constitutes a Bernoulli process. Interest centers on X, the number of units among these seven that need additional adjustments. The mean and variance of X are, respectively
    1. .15; .85
    2. .15; 1.05
    3. .15; .8925
    4. 1.05; .1275
    5. 1.05; .8915
    Solution e - remember variance = (std dev) squared
12. If you buy one ticket in the Provincial Lottery, then the probability that you will win a prize is 0.11. If you buy one ticket each month for five months, what is the probability that you will win at least one prize?
    1. 0.55
    2. 0.50
    3. 0.44
    4. 0.45
    5. 0.56
    Solution c
13. Suppose that the probability that a cross between two varieties will express a particular gene is 0.20. What is the probability that in 8 progeny plants, two or fewer plants will express the gene?
    1. .2936
    2. .3355
    3. .1678
    4. .6291
    5. .7969
    Solution e Past performance - 89 Oct - 95%
14. Refer to the previous question. Suppose that 120 crosses are bred. Find a likely 95% range for the number of progeny that will express the gene.
    1. 24±19.2
    2. 24±4.4
    3. 24±8.8
    4. 24±4.9
    5. 24±9.8
    Solution c Past performance - 89 Oct - 65%
15. Seventeen people have been exposed to a particular disease. Each one independently has a 40% chance of contracting the disease. A hospital has the capacity to handle 10 cases of the disease. What is the probability that the hospital's capacity will be exceeded?
    1. .965
    2. .035
    3. .989
    4. .011
    5. .736
    Solution b Past performance - 91 Oct - 75% Past performance - 93 Feb - 59% (c-14%; d-14%) Past performance - 93 Apr - 70% Past performance - 96 Nov - 90%
16. Refer to the previous problem. Planners need to have enough beds available to handle a proportion of all outbreaks. Suppose a typical outbreak has 100 people exposed, each with a 40% chance of coming down with the disease. Which is not correct:
    1. This experiment satisfies the assumptions of a binomial distribution.
    2. About 95% of the time, between 30 and 50 people will contract the disease.
    3. Almost all of the time, between 25 and 55 people will contract the disease.
    4. On average, about 40 people will contract the disease.
    5. Almost all of time, less than 40 people will be infected.
    Solution e Past performance - 93 Feb - 73% (d-13%) Past performance - 96 Nov - 80% (d- 8%)
17. There are 10 patients on the Neo-Natal Ward of a local hospital who are monitored by 2 staff members. If the probability (at any one time) of a patient requiring emergency attention by a staff member is .3, assuming the patients to be behave independently, what is the probability at any one time that there will not be sufficient staff to attend all emergencies?
    1. .3828
    2. .3000
    3. .0900
    4. .9100
    5. .6172
    Solution e

18. A newborn baby whose Apgar score is over 6 is classified as normal and this happens in 80% of births. As a quality control check, an auditor examined the records of 100 births. He would be suspicious if the number of normal births in the sample of 100 births fell above the upper limit of a "95%-normal-range". What is this upper limit?
    1. 112
    2. 72
    3. 88
    4. 8
    5. none of these
    Solution c Past performance - 73% (18% -e)

19. Refer to the previous question. Babies that have Apgar scores of 6 or lower require more expensive medical care. What is the probability that in the next 10 births, 3 or more babies will have Apgar scores of 6 or lower?
    1. .2013
    2. .3222
    3. .9999
    4. .0001
    5. .1536
    Solution c Past performance - 48% (19%-c; 11%-d; 14%-e)

These questions are from the Simon Fraser University statistics program web pages and have been captured here to make it easier to use. Thanks to the original authors..