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GATE Overflow | Mock GATE | Test 1 | Question: 23

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What is the minimum number of people that must be there in a room to make the probability of two people having same birthday be at least 50%? Assume  a year has $365$ days and the probability distribution is uniform throughout.

  1. $23$
  2. $182$
  3. $183$
  4. $123$
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If there are $n$ people in the room probably that no one share their birthday $ = \frac{365.364.363. \dots (365-n+1)}{365^n}$

We want this to be $\leq 0.5$ so that probability that at least $2$ share their birthday becomes $\geq 0.5$

i.e., we want least $n$ such that $$\frac{365.364.363. \dots (365-n+1)}{365^n} \leq 0.5$$

This happens for $n = 23.$
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