The Binomial Distribution:
The main part of a binomial experiment is to find the probability of r successes
out of n trials. (Think of a multiple choice exam that you didn’t study for. What are you
possible outcomes?)
Table 5 – 1 (text p. 269) shows you the possible outcomes for a binomial experiment with
3 trials.
*** View Guided exercise #4 p. 270.
General Formula for Binomial Probability Distribution
P(r) = Cn,rprqn – r
Where Cn,r = (n!)/(r!(n – r)!) is the binomial coefficient. Values of Cn,r for various values
of n and r can be found in Table 3 of Appendix II.
For each question on binomial distribution find:
1) Number of trials (n).
2) Number of successes (r).
3) Probability of success (p).
4) Probability of failure (q).
Then, just use the formula from above.
*** Try Example #3 text p. 272 – 273.
*** View Guided Exercise text p. 274 – 275.
The calculator can do this formula for you, just know what values to put in. Under the
2nd DISTR menu you will find the commands for finding Binomial probabilities.
For example: Select binompdf, then after the parenthesis put in n, p, r to get the answer.
Examples:
1) The local police department claims to solve 90% of all crimes committed in their
jurisdiction. The local newspaper reported that there have been 54 crimes in the past two
months, of which 46 have been solved. We would like to know what is the probability
that 46 crimes have been solved.
a) What makes up a trial?
b) What is a success?
c) What is a failure?
d) What is the value of n, p, q, and r?
2) A random sample of 58 people watching TV showed that 18 preferred sports, 21
preferred talk shows, 10 preferred late night movies, and 9 preferred murder mystery
shows. It is claimed that of the people who watch TV, 30% prefer sports, 36% prefer talk
shows, 17% prefer late night movies, and 17% prefer murder mysteries. Could we use
the binomial distribution to check the correctness of the claim? Give a reason for your
answer.
3) The college health center did a campus-wide survey of students and found that 15%
smoke. A group of nine students randomly come together in the student cafeteria and sit
at the same table. Assume the cafeteria does not have a nonsmoking section and that a
student who ordinarily smokes will smoke at this time. Find the probability that:
a) no student at this table smokes.
b) at least one student smokes.
c) more than two students smoke.
d) from one to five smoke (including one and five).
4) The mayor said that the local fire stations are so well equipped and staffed that they
are able to get to a fire in five minutes or less about 80% of the time. If this is so, what is
the probability that out of the next ten fire calls:
a) the fire department arrives in five minutes or less for each call?
b) for at least three calls it takes more than five minutes for the fire department to
arrive?
c) it takes more than five minutes for the fire department to arrive for all ten calls?
d) it takes five or less minutes for half of the calls?
5) A TV sports commentator claims that 45% of all football injuries are knee injuries.
Assuming that this claim is true, what is the probability that in a game with five reported
injuries
a) all are knee injuries?
b) none are knee injuries?
c) at least three are knee injuries?
d) no more than two are knee injuries?
6) A safety engineer claims that 20% of all automobile accidents are due to mechanical
failure. If this is correct, what is the probability that out of eight automobile accidents,
exactly three are due to mechanical failure?
7) In Summit County 65% of the voter population are Republicans. What is the
probability that a random sample of ten Summit County voters will contain
a) exactly eight Republicans?
b) exactly two Republicans?
8) Only three out of four patients who have are artery bypass heart operation are known
to survive five years without needing another bypass operation. Of seven patients who
recently had such an operation, what is the probability that
a) all will survive five years without needing another operation?
b) at least four will survive five years without needing another operation?
c) exactly four will survive five years without needing another operation?