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Discrete Math
Probability, Part 1 Exercises
1. In a small community called Broskers, someone examined the number of people
within various age groups that ride motorcycles. They summarized the data as
follows:
MC Rider
Non-rider
Total
18-25
80
420
500
26-40
120
580
700
41-60
80
720
800
Total
280
1,720
a. If you select a member of the community at random, what is the probability
that he or she will be a motorcycle rider?
b. What is the probability of selecting a motorcycle rider at random from the
entire group of 26-40 year olds?
c. What is the probability of selecting someone who is 26-40 from among all
motorcycle riders?
d. Are the events “Motorcycle Rider” and “Age 26-40” independent? How do
you know?
2. At South High School, the number of students in grades 10-12 that own a laptop
are as follows:
Sophomores
Juniors
Seniors
Total
Owns Laptop
40
40
40
120
No Laptop
180
160
140
480
Total
220
200
180
600
a. What is the probability that a student selected at random will own a laptop?
b. From among the Juniors, what is the probability of randomly selecting a
laptop owner?
c. Are being a Junior and owning a laptop independent? How do you know?
3. Last year, the Hooskins Hounds basketball team was 12-3 at home, and 8-7 on the
road. Are the events “Winning” and “Winning at Home” independent? Explain.
How does your answer fit with what you have witnessed in sports?
4. A bag contains 4 red marbles, 4 green marbles, and 2 blue marbles. You will
reach into the bag without looking and take out a marble. You record the color of
the marble, replace it in the bag, and then repeat. What is the probability that:
a. the first marble will be blue and the second marble will be red?
b. both marbles will be blue?
c. one (and only one) of the marbles will be green?
5. Using the same bag, this time after you take out one marble, you do not replace it.
Without replacement, what is the probability that:
a. the first marble will be blue and the second marble will be red?
b. both marbles will be blue?
c. one (and only one) of the marbles will be green?
6. Recall that a standard deck of cards has 52 cards. There are 4 suits; hearts and
diamonds are red; spades and clubs are black. Each suit contains the numbers 210, a Jack, a Queen, a King, and an Ace. You draw one card from the deck at
random, replace it, and then draw a card at random again. What is the probability
that:
a.
b.
c.
d.
e.
f.
both cards will be red?
the first card will be red and the second card will be black?
both cards will be a Queen?
you will draw a Queen first and then an Ace?
you will draw one Queen and one Ace, in either order?
you will draw two cards of the same suit?
7. This time you draw one card, and then without replacement draw a second card.
What is the probability that:
a.
b.
c.
d.
e.
f.
both cards will be red?
the first card will be red and the second card will be black?
both cards will be a Queen?
you will draw a Queen first and then an Ace?
you will draw one Queen and one Ace, in either order?
you will draw two cards of the same suit?
8. In the city of Boietporez, it has been found that when it rains one day, 60% of the
time it also rains the following day. If it does not rain one day, 30% of the time it
rains the following day. On the news it was announced that the probability of rain
today is 70%.
a. What is the probability that it will rain both today and tomorrow?
b. What is the probability that it will rain today but not tomorrow?
c. What is the probability that it will rain either today or tomorrow, but not
both days?
d. What is the probability that it will rain neither today nor tomorrow?
e. Is tomorrow’s weather independent of today’s weather? Explain.
9. The test for a certain disease has been determined to be 99% accurate. In other
words, of those who test positive, 99% actually have the disease. Of those who
test negative, 99% do not have the disease. It is estimated that 4 out of every
1,000 people has the disease.
Disease
Tests Positive
Tests Negative
Tests Positive
No
Disease
Tests Negative
a. Calculate each combined probability (e.g., having the disease and testing
positive) and record it in the appropriate space on the right of the tree
diagram.
b. For every 100,000 people that take the test, how many would you predict
would test positive?
c. Out of those that test positive in part b, how many would you predict
actually have the disease?
10. If you flip a coin 4 times, what is the probability that you will get Heads all 4
times?
11. A certain machine in a factory contains a series of 6 gears. If any one of the gears
fails, the machine will not operate properly.
a. If each gear has a 99% probability of working individually, what is the
probability that all 6 will be working properly at any given time?
b. The company is considering purchasing some less expensive gears. These
gears have a 97% probability of working individually. What is the
probability that all 6 would function properly?
12. For rolling a pair of dice (one red die and one green die):
a. How many total outcomes are possible for one roll of both dice?
b. What is the probability of getting doubles?
c. What is the probability of rolling doubles twice in a row?