Concept explainers
Suppose that a six-sided die is weighted so that any even-numbered face is twice as likely to land face up as any odd-numbered face. Consider the chance experiment that consists of rolling this die.
- a. What are the probabilities of the six simple events? (Hint: Denote these events by O1, …, O6. Then P(O1) = p, P(O2) = 2p, P(O3) = p, …, P(O6) = 2p. Now use a condition on the sum of these probabilities to determine p.)
- b. What is the
probability that the number showing is an odd number? What is the probability that the number showing is at most three? - c. Now suppose that the die is weighted so that the probability of each simple
event is proportional to the number showing on the corresponding upturned face; that is, P(O1) = c, P(O2) = 2c, …, P(O6) = 6c. What are the probabilities of the six simple events? Calculate the probabilities of Part (b) for this die.
a.
Calculate the probabilities of the simple events.
Answer to Problem 37E
The probabilities of the simple events are given by:
Explanation of Solution
Calculation:
It is given that even-numbered face is twice as likely to occur as odd-numbered face.
Events
Therefore, the probability of sample events is given by:
Thus, the probabilities of the sample events are given by:
b.
Calculate the probability of getting an odd number when a die is rolled.
Calculate the probability that number showing on the die is at most three.
Answer to Problem 37E
The probability of getting an odd number when a die is rolled is
The probability that number showing on the die is at most three is
Explanation of Solution
Calculation:
The required probability is given by:
From part (a), it is clear that
Therefore, the required probability is given by:
Thus, the probability of getting an odd number when a die is rolled is
Therefore, the required probability is given by:
Thus, the probability that number showing on the die is at most three is
c.
Calculate the probabilities of the simple events.
Calculate the probabilities of part (b).
Answer to Problem 37E
The probabilities of the simple events are given by:
The probability of getting an odd number when a die is rolled is
The probability that number showing on the die is at most three is
Explanation of Solution
Calculation:
It is given that even-numbered face is twice as likely to occur as odd-numbered face.
Events
Therefore, the probabilities of sample events are given by:
Thus, the probabilities of the simple events are given by:
The required probabilities are obtained as follows:
Thus, the probability of getting an odd number when a die is rolled is
Thus, the probability that number showing on the die is at most three is
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Chapter 6 Solutions
Introduction To Statistics And Data Analysis
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