Problems: Probability Rules
1. In 17th century France gamblers bet on whether at
least one ace would show up in four rolls of a die. However,
they did not know how to calculate that probability. Two
mathematicians decided to solve the problem by computing the
probability that no aces would turn up in four rolls. They
got an answer of a little more than 48%.
 a) How do you compute the probability that no aces
will show in four rolls?
 b) Given that you know that the probability of
getting no aces is 48%, can you figure out what the
probability of getting exactly one ace? Explain.
 c) Given that you know that the probability of
getting no aces is 48%, can you figure out what the
probability of getting at least one ace? Explain.
2a) Two cards are drawn from a deck of cards without
replacement. What is the probability that the first will be
a red card (a heart or diamond) and the second will be a
black card (a club or spade)?
b) Two cards are drawn from a deck of cards with
replacement. What is the probability that the first will be
a red card (a heart or diamond) and the second will be a
black card (a club or spade)?
c) In which case, a or b (or either or both) do we have
independence?
3a) What is the probability of drawing two consecutive
face cards from a 52card deck? (There are 12 face cards in
a deck; the first card is not replaced before the second is
drawn.)
b) A deck is cut twice. What is the probability that both
cuts will reveal a face card? (No card is removed when the
deck is cut.)
c) What is the probability of drawing a face card or a
diamond if one card is drawn from a 52card deck?
d) What is the probability of drawing both a face card and a
diamond?
e) Given that you have drawn a face card, what is the
probability that you have drawn a queen?
4. Suppose that events A and B are mutually exclusive.
Can they also be independent? Explain.
5. A family is planning a picnic for Labor Day. The will
not go if it rains or if the father can work that day.
(Overtime pays well.) The probability that it will rain is
.15 and the probability that the father will be able to work
is .2 Assuming that the two events are independent, what is
the probability that they will go on the picnic?
6. A sportsman is planning a hunting trip but will not go
if there is rain or if he can work on the planned day. He
figures that probability of rain to be .25 and the
probability of working that day to be .30. He also decides
that the probability that he will be able to go on the trip
is .45.
 a) Based on these probabilities, are the events
"rain" and "work" collectively exhaustive?
 b) Are the events "rain" and "work" mutually
exclusive? Explain.
 c) Are the events "rain" and "work" independent?
Explain.
7. A statistics instructor figures that following
probabilities: if a student reads his text, there is an 80%
chance that he will get a grade of C or better, and if he
does not read the book, there is a 30% chance that he will
get a C or above. From past experience the instructor knows
that only 70% of his class is likely to read the text. (This
question is purely hypothetical.) What is the probability
that a student chosen at random will get a D or F?
8. Ms Jones has two children. Given that at least one of
them is a girl, what is the probability that the other is a
boy? (Hint: do not rely on your intuition.)
9. From the five finalists in the Miss America contest,
one will be selected as the winner and one as the first
runnerup. How many different arrangements of winner and
first runnerup are possible?
10. A Russian Roulette player has played his game eight
times and is still alive.
 a) If the probability of losing is 1/6, find the
exact probability of winning this game eight times
straight.
 b) How many times do you have to play this game
before the probability of losing once is greater than
50%
11. Suppose you had a loaded die (singular of dice) that
will come up with an ace (one) one third of the time. One
twelfth of the time it will come up with two spots showing,
and the same chance of one twelfth exists for each of 3, 4,
and 5.
 a) What is the probability of rolling a six with this
die?
 b) Suppose you roll this die twice. What is the
probability of rolling a two on the first throw and then
a one?
Answers here.
More problems
12. I have a package of tomato seeds. 60% of them are a
variety of yellow tomatoes and 40% of them are a variety of
red tomatoes. The germination rate is 80%. What is the
probability that if I select one seed at random, I will end
up eating big red tomatoes this fall? (Hint &endash; the
probability that a seed will be both yellow and will not
germinate is .08. Use your laws of probability, and start by
finding the probability that I will not get red
tomatoes.)
13. In a certain community, the probability that a family
has a television set is .80, a washing machine .50, both a
television set and a washing machine is 0.45. What is the
probability that a family has either a television set or a
washing machine or both?
14 In Littletown 70% of the households have a pet dog,
50% have a pet cat, and 40% have both a pet dog and a pet
cat. What is the probability that a household chosen at
random will have neither a pet cat nor a pet dog?
15. A die has six sides numbered one through six and each
side has a onesixth chance of landing on top when the die
is tossed. When you throw the die,
 a) what is the probability that a one comes up? What
is the probability of "not one"?
 b) what is the probability that either a one or a two
come up? What is the probability of "not one or
two"?
 c) what is the probability that a number three or
less will come up?
 d) what is the probability that an odd number will
come up?
 e) what is the probability that either an odd number
or a number three or less will come up?
 f) what is the probability of getting that both an
odd number and a number three or less?
 g) what is the probability that an odd number will
come up given that the result is three or less?
