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 52-card 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 52-card 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 runner-up. How many different arrangements of winner and first runner-up 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.

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 one-sixth 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?

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