Bunuel wrote:
{x, y, 1, 2, 3}
If x and y are positive integers less than 10, what is the median of the list above?
(1) When one number is chosen at random from the list, the probability of selecting a multiple of 2 is less than the probability of selecting a non-prime number
(2) When one number is chosen at random from the list, the probability of selecting a multiple of 3 is greater than the probability of selecting a prime number
Official Solution:\(\{x, \ y, \ 1, \ 2, \ 3\}\) If \(x\) and \(y\) are positive integers less than 10, what is the median of the list above? Recall that the median of a list is the middle term, when arranged in ascending or descending order.
(1) When one number is chosen at random from the list, the probability of selecting a multiple of 2 is less than the probability of selecting a non-prime number.
The above means that there are more non-prime numbers in the list than multiples of 2. Since without \(x\) and \(y\) there are equal numbers of non-prime numbers and multiples of 2 in the list (non-prime is 1 and multiple of two is 2), then in \(\{x, \ y \}\) must be more non-primes than multiples of 2.
We could have several different cases. For example, if \(\{x, \ y\} =\{1, \ 1\}\), the the list would be \(\{1, \ 1, \ 1, \ 2, \ 3 \}\) and the median would be 1 but if \(\{x, \ y\}=\{9, \ 9 \}\), the the list would be \(\{1, \ 2, \ 3, \ 9, \ 9 \}\) and the median would be 9. Two different answers. Not sufficient.
(2) When one number is chosen at random from the list, the probability of selecting a multiple of 3 is greater than the probability of selecting a prime number.
The above means that there are more multiples of 3 in the list than primes. Since without \(x\) and \(y\) there are less multiples of 3 than primes (multiple of 3 is 3 and primes are 2, and 3), then both \(x\) and \(y\) must be multiples of 3 but not primes. So, each of \(x\) and \(y\) could be 6 or 9. Now, if both \(x\) and \(y\) are more that 3 (6 or 9), then the list in ascending order would be \(\{1, \ 2, \ 3, x, \ y\}\), which means that the median is 3. Sufficient.
Answer: B
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