Riva – Adv Precalc
Conditional Probability and Binomial Distribution Problems
1) Find 𝑃(𝐴|𝐵) and 𝑃(𝐵|𝐴) given that P(A)=0.7, P(B)=0.4, and 𝑃(𝐴 ∩ 𝐵) = 0.25
2) Find P(A) and P(B) given that 𝑃(𝐴 ∩ 𝐵) = 0.2, 𝑃(𝐴|𝐵) = 0.54, and
𝑃(𝐵|𝐴) = 0.85
3) Use the data in the table, which shows the employment status of individuals
in a particular town by age group, to answer the following questions
4)
5)
6)
7)
8)
a. If a person in this town is selected at random, find the probability that
the individual is employed part-time, given that he or she is between
the ages of 35 and 49.
b. If a person in the town is randomly selected, what is the probability
that the person is unemployed, given that he or she at least 35 years
old.
c. A person from the town is randomly selected. What is the probability
that the individual is employed part-time, given that he or she is
under 25 years old?
The events, A and B, are independent if 𝑃(𝐴|𝐵) = 𝑃(𝐴) or 𝑃(𝐵|𝐴) = 𝑃(𝐵).
Determine whether the following events are independent. Then, find
𝑃(𝐴 ∩ 𝐵).
a. In order to determine who goes first in a game, one person picks a
number between 1 and 10, then a second person picks a number
between 1 an 10. A={the sum of the numbers is even} and B={the first
number is a 4}.
b. A fair coin is tossed four times in succession. A={the first two tosses
are heads} and B={the last two tosses are tails}
A standard deck of playing cards is shuffled and three people each choose a
card. Find the probability that the first two cards chosen are diamonds and
the third card is red if
a. the cards are chosen with replacement.
b. the cards are chosen without replacement
A standard deck of playing cards is shuffled and four people each choose a
card. Find the probability that the first two cards chosen are kings, the third
card is black, and the fourth card is a spade if
a. the cards are chosen with replacement.
b. the cards are chosen without replacement
A company owns 400 laptops. Each laptop has an 8% probability of not
working. You randomly select 20 laptops for your salespeople.
a. What is the likelihood that 5 will be broken?
b. What is the probability that they will all work?
c. What is the likelihood that at least two will be broken?
An HDTV is made from 100 components. Each component has a 0.005
probability of being defective. What is the probability that an HDTV will not
work perfectly?
Riva – Adv Precalc
9) The probability that a student enrolled at a local high school will be absent
on a particular day is 0.04, assuming that the student was in attendance the
previous school day. However, if a student is absent, the probability that he
or she will be absent again the following day is 0.11. For each exercise,
assume that the student was in attendance the previous day.
a. What is the probability that a student will be absent three days in a
row?
b. What is the probability that a student will be absent two days in a
row, but then show up on the third day?
c. Find the probability that a student will be absent, attend the next day,
but then will be absent again the third day.
d. Find the probability that a student will be absent four days in a row.
10)Suppose that a company claims it has a test that is 95% effective in
determining whether an athlete is using a steroid. That is, if an athlete is
using a steroid, the test will be positive 95% of the time. In the case of a
negative result, the company says its test is 97% accurate. That is, even if an
athlete is not using steroids, it is possible that the test will be positive in 3%
of the cases. Such an occurrence is called a false positive. Suppose this test is
given to a group of athletes in which 10% of the athletes are using steroids.
a. What is the probability that a randomly chosen athlete actually uses
steroids, given that the athlete's test is positive?
b. What is the probability that a randomly chosen athlete does not use
steroids, given that the athlete's test is positive?
c. What is the probability that a randomly chosen athlete actually uses
steroids, given that the athlete's test is negative?
d. What is the probability that a randomly chosen athlete does not use
steroids, given that the athlete's test is negative?
11)A company that performs drug testing guarantees that its test determines a
positive result with 97% accuracy. However, the test also gives 6% false
positives. If 5% of those being tested actually have the drug present in their
bloodstream, find the following probabilities:
a. What is the probability that a randomly chosen person actually has
the drug present in their bloodstream, given that the test is positive?
b. What is the probability that a randomly chosen person does not have
the drug present in their bloodstream, given that the test is positive?
c. What is the probability that a randomly chosen person actually has
the drug present in their bloodstream, given that the test is negative?
d. What is the probability that a randomly chosen person does not have
the drug present in their bloodstream, given that the test is negative?
12)A test for a genetic disorder can detect the disorder with 94% accuracy.
However, the test will incorrectly report positive results for 3% of those
without the disorder. 12% of the population has the disorder.
a. What is the probability that a randomly chosen person actually has
the genetic disorder, given that the test is positive?
b. What is the probability that a randomly chosen person does not have
the genetic disorder, given that the test is positive?
c. What is the probability that a randomly chosen person actually has
the genetic disorder, given that the test is negative?
d. What is the probability that a randomly chosen person does not have
the genetic disorder, given that the test is negative?
Riva – Adv Precalc
13)A pharmaceutical company has developed a test for a scarce disease that is
present in 0.5% of the population. The test is 98% accurate in determining a
positive result, and the chance of a false positive is 4%.
a. What is the probability that a randomly chosen person actually has
the disease, given that the test is positive?
b. What is the probability that a randomly chosen person does not have
the disease, given that the test is positive?
c. What is the probability that a randomly chosen person actually has
the disease, given that the test is negative?
d. What is the probability that a randomly chosen person does not have
the disease, given that the test is negative?
14)The Centers for Disease Control and Preventi6n estimate that about 0.3% of
U.S. residents are infected with HIV. When used together, the ELISA and
Western Blot tests for HIV are more than 99% accurate in determining a
positive result. The chance of a false positive is between 1 and 5 for every
100,000 tests. If we assume a 99% accuracy rate for correctly identifying
positive results and 5/100,000 false positives.
a. What is the probability that a randomly chosen personactually has
HIV, given that the test is positive?
b. What is the probability that a randomly chosen person does not have
HIV, given that the test is positive?
c. What is the probability that a randomly chosen person actually has
HTV, given that the test is negative?
d. What is the probability that a randomly chosen person does not have
HTV, given that the test is negative?
15)A study indicates that 4% of American teenagers have tattoos. You randomly
sample 30 teenagers. What is the likelihood that exactly three will have
tattoos?
16)A certain brand of cell phone is made from 55 components. Each component
has a 0.002 probability of being defective. What is the probability that a given
cell phone will not work perfectly?
17)The ABC company manufactures toy robots. About 1 toy robot in 100 does
not work. You purchase 35 AMC toy robots. What is the probability that
exactly 4 do not work?
18)The LMB Company manufactures tires. They claim that only 0.007 of LMB
tires are defective. What is the probability of finding 2 defective tires in a
random sample of 50 LMB tires?
Riva – Adv Precalc
Conditional Probability and Binomial Distribution Problems Answers
5
5
1) 𝑃(𝐴|𝐵) = ; 𝑃(𝐵|𝐴) =
4
8
10
14
2) 𝑃(𝐴) = 17; 𝑃(𝐵) = 27
179
3) a) 864
277
b) 1642
367
c) 1095
1
4) a) independent; 𝑃(𝐴 ∩ 𝐵) = 12
1
1
b) independent; 𝑃(𝐴 ∩ 𝐵) = 16
12
5) a) 32
b) 425
6) a) 1352
7) a) .0145
8) .3942
9) a) .000484
95
10)a) 122
b) 6497400
b) .1887
c) .4831
b) .003916
27
b) 122
c) .001424
5
c) 878
d) .00005324
873
d) 878
11)a) 211
b) 211
c) 1789
d) 1789
12)a) 58
b) 58
c) 1076
d) 1076
13)a) 447
14)a) .9835
15) .0863
16).1043
17).000384
18).0428
b) 447
b) .0165
c) 9953
c) .00003
d) 9953
d) .99997
1
97
47
49
3602
114
11
398
3
9
1
1786
1067
9952