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31 Dec 2023, 10:15
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Elvin has sent wish lists to both Santa and Mrs. Claus. What is the probability that he will receive gifts from neither of them?
(1) The probability that he will receive gifts from both Santa and Mrs. Claus is \(\frac{1}{4}\).
(2) The probability that he will receive gifts from exactly one of them is three times the probability that he will receive gifts from both.
(1) The probability that he will receive gifts from both Santa and Mrs. Claus is \(\frac{1}{4}\).
(2) The probability that he will receive gifts from exactly one of them is three times the probability that he will receive gifts from both.
31 Dec 2023, 10:15
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Official Solution:
Elvin has sent wish lists to both Santa and Mrs. Claus. What is the probability that he will receive gifts from neither of them?
We have:
{Santa only} + {Mrs. Claus only} + {Both} + {Neither} = 1
The question asks to find the probability of {Neither}.
(1) The probability that he will receive gifts from both Santa and Mrs. Claus is \(\frac{1}{4}\).
This implies that {Both} = \(\frac{1}{4}\). Not sufficient.
(2) The probability that he will receive gifts from exactly one of them is three times the probability that he will receive gifts from both.
This implies that {Santa only} + {Mrs. Claus only} = 3*{Both}. Not sufficient.
(1)+(2) Since from (1) {Both} = \(\frac{1}{4}\) and from (2) {Santa only} + {Mrs. Claus only} = 3*{Both}, then {Santa only} + {Mrs. Claus only} = \(\frac{3}{4}\). Thus, we have that \(\frac{3}{4} + \frac{1}{4} + \text{{Neither}} = 1\), which yields {Neither} = 0. Of course, Elvin won't be left without gifts on Christmas! Sufficient.
Answer: C
Elvin has sent wish lists to both Santa and Mrs. Claus. What is the probability that he will receive gifts from neither of them?
We have:
{Santa only} + {Mrs. Claus only} + {Both} + {Neither} = 1
The question asks to find the probability of {Neither}.
(1) The probability that he will receive gifts from both Santa and Mrs. Claus is \(\frac{1}{4}\).
This implies that {Both} = \(\frac{1}{4}\). Not sufficient.
(2) The probability that he will receive gifts from exactly one of them is three times the probability that he will receive gifts from both.
This implies that {Santa only} + {Mrs. Claus only} = 3*{Both}. Not sufficient.
(1)+(2) Since from (1) {Both} = \(\frac{1}{4}\) and from (2) {Santa only} + {Mrs. Claus only} = 3*{Both}, then {Santa only} + {Mrs. Claus only} = \(\frac{3}{4}\). Thus, we have that \(\frac{3}{4} + \frac{1}{4} + \text{{Neither}} = 1\), which yields {Neither} = 0. Of course, Elvin won't be left without gifts on Christmas! Sufficient.
Answer: C
General Discussion
03 Jun 2025, 21:56
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