Section 4.6 Probability
315
S e c t i o n 4 . 6 Review
techniqueS
• Compute the probability of an event when all outcomes of an action are equally likely.
• Compute the probability of an event when a probability distribution has been assigned to the sample
space.
• Compute the conditional probability of an event
given that another event has already occurred.
• Determine whether two events are independent.
• Given a sample space with a random variable x and
a probability distribution, compute the expected
value E(X ).
Main ideaS
• An event is a subset of the set of all possible outcomes of some action.
• For equally likely outcomes, the probability of an
event is the ratio of the number of outcomes in the
event to the number of all possible outcomes.
• In the conditional probability of event E 2 given that
event E 1 has already taken place, the sample space
is reduced to E 1 .
• Events E 1 and E 2 are independent if and only if the
conditional probability of E 2 given E 1 is the same
as the probability of E 2 .
• In the simplest form of Bayes’ theorem, the probability of event A given event B can be computed
from the probability of A, the probability of B, and
the probability of B given A.
• Given a sample space to which a random variable
and a probability distribution have been assigned,
the expected value of the random variable is a predictor of its future value.
• An average case analysis of an algorithm is the
expected value of work units over the sample
space of all inputs; the probability distribution reflects the assumptions being made about “ average”
input.
W
W
exerciSeS 4.6
Exercises 1−6 concern three coins tossed at the same time, each equally likely to come up heads or tails.
1. What is the size of the sample space?
2. What is the probability of getting 1 head and 2 tails?
3. What is the probability of getting all tails?
4. What is the probability that no coin comes up heads?
5. What is the probability of getting all tails or all heads?
6. What is the probability of getting all tails and all heads?
In Exercises 7−14, a pair of fair dice is rolled.
7. What is the size of the sample space?
8. What is the probability of getting “snake eyes” (two 1s)?
9. What is the probability of getting doubles (the same number on each die)?
10. What is the probability of getting a 1 on at least one die?
11. What is the probability of getting a total of 7 on the two dice?
12. What is the probability of getting two consecutive values, such as 3−4, on the two dice?
13. What is the probability of getting a total on the two dice greater than 10?
14. What is the probability of getting a total on the two dice that is an odd number?
315
S e c t i o n 4 . 6 Review
techniqueS
• Compute the probability of an event when all outcomes of an action are equally likely.
• Compute the probability of an event when a probability distribution has been assigned to the sample
space.
• Compute the conditional probability of an event
given that another event has already occurred.
• Determine whether two events are independent.
• Given a sample space with a random variable x and
a probability distribution, compute the expected
value E(X ).
Main ideaS
• An event is a subset of the set of all possible outcomes of some action.
• For equally likely outcomes, the probability of an
event is the ratio of the number of outcomes in the
event to the number of all possible outcomes.
• In the conditional probability of event E 2 given that
event E 1 has already taken place, the sample space
is reduced to E 1 .
• Events E 1 and E 2 are independent if and only if the
conditional probability of E 2 given E 1 is the same
as the probability of E 2 .
• In the simplest form of Bayes’ theorem, the probability of event A given event B can be computed
from the probability of A, the probability of B, and
the probability of B given A.
• Given a sample space to which a random variable
and a probability distribution have been assigned,
the expected value of the random variable is a predictor of its future value.
• An average case analysis of an algorithm is the
expected value of work units over the sample
space of all inputs; the probability distribution reflects the assumptions being made about “ average”
input.
W
W
exerciSeS 4.6
Exercises 1−6 concern three coins tossed at the same time, each equally likely to come up heads or tails.
1. What is the size of the sample space?
2. What is the probability of getting 1 head and 2 tails?
3. What is the probability of getting all tails?
4. What is the probability that no coin comes up heads?
5. What is the probability of getting all tails or all heads?
6. What is the probability of getting all tails and all heads?
In Exercises 7−14, a pair of fair dice is rolled.
7. What is the size of the sample space?
8. What is the probability of getting “snake eyes” (two 1s)?
9. What is the probability of getting doubles (the same number on each die)?
10. What is the probability of getting a 1 on at least one die?
11. What is the probability of getting a total of 7 on the two dice?
12. What is the probability of getting two consecutive values, such as 3−4, on the two dice?
13. What is the probability of getting a total on the two dice greater than 10?
14. What is the probability of getting a total on the two dice that is an odd number?