Лекция: Exercises (using independent events)

1 In a mathematics test the probability that Aly scores more than 70% is 0.6. In a physics test the probability that he scores more than 70% is 0.5. What is the probability that

a) he scores more than 70% in both tests

b) he scores more than 70% in only one test?

2 A cubical die is thrown twice.

a) Draw a tree diagram to show the outcomes “throwing a three” and “not throwing a three”.

b) What is the probability that both dice show a three?

c) What is the probability that neither dice shows a three?

d) Draw a tree diagram to show the outcomes “throwing a number less than four” and “throwing a number greater than or equal to four”.

e) What is the probability that only one die shows a number less than four?

f) What is the probability that at least one die shows a number less than four?

3 On any particular day, the probability that it rains is 0.2. The probability that a soccer team will win is 0.6 if it is raining and 0.7 if it is not raining. The team plays once in a week.

a) Draw a tree diagram to show these events and their outcomes.

b) What is the probability that it will rain and the team will win?

c) What is the probability that the team will lose?

d) Given that it is not raining, what is the probability that they will lose?

e) Given that they win, what is the probability that it was raining?

4 Three fair coins are tossed. Each coin can either land on a head or a tail. What is the probability of gaining

a) three heads

b) two heads and a tail

c) a tail on the first toss followed by a head or a tail in either order on the second toss

d) at least one tail?

5 Two fair coins are tossed. Each coin can either land on a head or a tail.

a) Show the possible outcomes on a tree diagram.

b) What is the probability of getting at least one head?

c) A third coin is now tossed which is twice as likely to show heads as tails.

d) Add an extra set of branches to the tree diagram to show the possible outcomes.

e) What is the probability of getting two heads and a tail?

f) What is the probability of getting two tails and a head, given that the third coin lands on tails?

6 The letters of the word PROBABILITY are placed in a bag. A letter is selected, it is noted whether it is a vowel or a consonant, and returned to the bag. A second letter is then selected and the same distinction is noted.

a) Draw a tree diagram to show the possible outcomes.

b) What is the probability of noting two consonants?

c) What is the probability of noting a vowel and a consonant?

7 A soccer player finds that when the weather is calm, the probability of him striking his target is 0.95. When the weather is windy, the probability of him striking his target is 0.65. According to the local weather forecast, the probability that any particular day is windy is 0.45.

a) Find the probability of him hitting the target on any randomly chosen day.

b) Given that he fails to hit the target, what is the probability that the day is calm?