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GATE Overflow Test Series | Probability | Test 1 | Question: 11

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Assume that there are at least $k$ different people in the room. If the probability that there are two who share a birthday is greater than $10\%$ then which of the following is the least possible value of $k$. (Assume no one is born in a leap year)

  1. $23$
  2. $12$
  3. $18$
  4. $10$
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P(shared birthday ) = 1 – P(no one share a birthday)

$\qquad= 1-\frac{n.(n-1)(n-2)\ldots (n-k+1)}{n^k}$

Here, $n= 365$ days, and we need to find $k$ such that probability exceed $0.1$

$\implies 0.1 \leq 1-\frac{365.364.363\ldots (365-k+1)}{365^k}$

$\implies \frac{365.364.363\ldots (365-k+1)}{365^k} \leq 0.9$

Trying the values, $k = 10.$
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