Concept explainers
Find the
Obtain the probability that at least one of the n individuals gets their own calculator.
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Answer to Problem 98SE
The probability that at least one of the five gets her own calculator is 0.633.
The probability that at least one of the n individuals gets their own calculator is
Explanation of Solution
Given info:
The information is based on five friends named Allison, Beth, Carol, Diane and Evelyn who has identical calculator. They placed the calculator together for a study break and after the break each pick one calculator at random.
Calculation:
Define the
Since, each student has an equal chance to pick a calculator, the events are equally likely.
The probabilities of the corresponding events are given below:
The different cases where two students pick their own calculator:
The number of occurrence that A and B get her own calculatoris
The number of occurrence that A andC her own calculator is
The number of occurrence that A andD her own calculator is
The number of occurrence that A andE her own calculator is
Similarly, the other combinations where two students pick their own calculator can be obtained.
The different cases where three students pick their own calculator:
The number of occurrence that A andB and C her own calculator is
Similarly, the other combinations where three students pick their own calculator can be obtained.
The different cases where four students pick their own calculator:
The number of occurrence that A and B and C and D her own calculator is
Similarly, the other combinations where four students pick their own calculator can be obtained.
The different cases where five students pick their own calculator:
The number of occurrence that A and B and C and D and E her own calculator is
Factorial of an integer:
The factorial of a non-negative integer n is given by
Substitute 5 for ‘n’
Thus, there are 120 ways that the 5 students pick an identical calculator.
The probability that A and B is obtained as:
The probability that A and C is obtained as:
The probability that A and D is obtained as:
The probability that A and E is obtained as:
The probability that B and C is obtained as:
The probability that B and D is obtained as:
The probability that B and E is obtained as:
The probability that C and D is obtained as:
The probability that C and E is obtained as:
The probability that C and D is obtained as:
The probability that D and E is obtained as:
The probability that A and B and C is obtained as:
The probability that A and B and D is obtained as:
The probability that A and B and E is obtained as:
The probability that A and C and D is obtained as:
The probability that A and C and E is obtained as:
The probability that B and C and D is obtained as:
The probability that B and C and E is obtained as:
The probability that C and D and E is obtained as:
The probability thatA andB and C and D is obtained as:
The probability that A and B and C and D and E is obtained as:
Addition rule:
For any five events A ,B, C,D and E
By using addition rule,
The probability that at least one of the five gets her own calculator is obtained as shown below:
Thus the probability that at least one of the 5individuals gets their own calculator is 0.633.
The probability that at least one of the n individuals gets their own calculator is obtained as shown below:
The probability that at least one of the 5individuals gets their own calculator can be expressed as
The above expression may also be written in the power series of
When n is large, there 63.2% for the at least one of the n individuals get their calculator. Rather in large group, there 36.8% chance that not at least one of the n individual get their calculator back.
The probability that at least one of the n individuals gets their own calculator is
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