Concept explainers
Rolling Die Two dice are rolled. Find the probability of getting
a. A sum of 8, 9, or 10
b. Doubles or a sum of 7
c. A sum greater than 9 or less than 4
d. Based on the answers to a, b, and c, which is least likely to occur?
a.
To obtain: The probability of getting a sum of 8, 9, or 10.
Answer to Problem 24E
The probability of getting a sum of 8, 9, or 10 is
Explanation of Solution
Calculation:
Mutually exclusive events:
The two events A and B are mutually exclusive events, then probability
The possibilities for rolling a pair of six-sided dice is,
Thus, the total number of outcomes is 36.
Let event A denote that the outcome as sum of 8. Hence, the possible outcomes to get the sum 8 are ‘
The formula for probability of event A is,
Substitute 5 for ‘Number of outcomes in A’ and 36 for‘Total number of outcomes in the sample space’,
Here, the event B is defined getting sum of 9.
Hence, the possible outcomes are ‘
The probability of event B is,
Substitute 4 for ‘Number of outcomes in B’ and 36 for ‘Total number of outcomes in the sample space’,
Here, the eventC is defined as getting a sum of 10. Hence, the possible outcomes for sum 10 are ‘
The probability of event C is,
Substitute 3 for ‘Number of outcomes in C’ and 36 for ‘Total number of outcomes in the sample space’,
Addition Rule:
Here, the events are mutually exclusive. Thus, the formula for finding the probability of getting event A or event B or event C is,
Substitute
Thus, the probability of getting a sum of 8, 9, or 10 is
b.
To obtain: The probability of getting doubles or a sum of 7.
Answer to Problem 24E
The probability of getting doubles or a sum of 7 is
Explanation of Solution
Calculation:
Let event D denote that the outcomes sum of 7. Hence, the possible outcomes to get the sum 7 are ‘
The formula for probability of event D is,
Substitute 6 for ‘Number of outcomes in A’ and 36 for ‘Total number of outcomes in the sample space’,
Let the event E denote getting doubles.
Hence, the possible outcomes are ‘
The formula for probability of event E is,
Substitute 6 for ‘Number of outcomes in D’ and 36 for ‘Total number of outcomes in the sample space’,
Addition Rule:
Here, the events are mutually exclusive. Thus, the formula for finding the probability of getting event E or event D,
Substitute
Thus, the probability of getting doubles or a sum of 7 is
c.
To obtain: The probability of getting sum greater than 9 or a less than 4.
Answer to Problem 24E
The probability of getting sum greater than 9 or a less than 4 is
Explanation of Solution
Calculation:
Let event A denote that the outcome as sum greater than 9. Hence, the possible outcomes to get the sum greater than 9 are ‘
The formula for probability of event A is,
Substitute 6 for ‘Number of outcomes in A’ and 36 for ‘Total number of outcomes in the sample space’,
Here, the event B is defined getting sum is less than 4.
Hence, the possible outcomes are ‘
The probability of event B is,
Substitute 3 for ‘Number of outcomes in B’ and36 for ‘Total number of outcomes in the sample space’,
Addition Rule:
Here, the events are mutually exclusive. Thus, the formula for finding the probability of getting event A or event B,
Substitute
Thus, the probability of getting sum greater than 9 or a less than 4 is
d.
To identify: The event which is least likely to occur.
Answer to Problem 24E
The event b is least likely to occur.
Explanation of Solution
From the probabilities of events a, b and c, it can be observed that the probability of event b is less the other probabilities of a and c. That is,
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