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19 Nov 2021, 04:22
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Minimum of how many people are needed to have the probability of more than 1/2 that at least one of them was born on either on Monday or on Tuesday?
A. \(2\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(6\)
A. \(2\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(6\)
19 Nov 2021, 04:22
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Official Solution:
Minimum of how many people are needed to have the probability of more than 1/2 that at least one of them was born on either on Monday or on Tuesday?
A. \(2\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(6\)
The probability that a person is born on either on Monday or on Tuesday is \(\frac{2}{7}\) and the probability that a person is NOT born on either on Monday or on Tuesday is therefore \(\frac{5}{7}\).
Say minimum \(n\) people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday. In this case the probability that NONE of them is born on either on Monday or on Tuesday will be \((\frac{5}{7})^n\). Thus, the probability that AT LEAST ONE of them is born on either on Monday or on Tuesday will be \(1-(\frac{5}{7})^n\).
The question asks to find minimum \(n\) such that \(1-(\frac{5}{7})^n > \frac{1}{2}\):
\( \frac{1}{2}>(\frac{5}{7})^n\);
The least value of \(n\) that satisfies this is 3: \((\frac{5}{7})^3=\frac{125}{343} < \frac{1}{2}\).
Answer: B
Minimum of how many people are needed to have the probability of more than 1/2 that at least one of them was born on either on Monday or on Tuesday?
A. \(2\)
B. \(3\)
C. \(4\)
D. \(5\)
E. \(6\)
The probability that a person is born on either on Monday or on Tuesday is \(\frac{2}{7}\) and the probability that a person is NOT born on either on Monday or on Tuesday is therefore \(\frac{5}{7}\).
Say minimum \(n\) people are needed to have the probability of more than 1/2 that at lease one of them was born on either on Monday or on Tuesday. In this case the probability that NONE of them is born on either on Monday or on Tuesday will be \((\frac{5}{7})^n\). Thus, the probability that AT LEAST ONE of them is born on either on Monday or on Tuesday will be \(1-(\frac{5}{7})^n\).
The question asks to find minimum \(n\) such that \(1-(\frac{5}{7})^n > \frac{1}{2}\):
\( \frac{1}{2}>(\frac{5}{7})^n\);
The least value of \(n\) that satisfies this is 3: \((\frac{5}{7})^3=\frac{125}{343} < \frac{1}{2}\).
Answer: B
19 May 2024, 17:17
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I think this is a high-quality question and I agree with explanation.
