Q.4.56 - A First Course in Probability

Question

How many people are needed so that the probability that at least one of them has the same birthday as you is greater than $\frac{1}{2}$ ? 

Step-by-Step Solution

Answer

The probability the same birthday found to be $253$.

Step 1: Given Information

The probability that at least one person has the same birthday as me is greater than $\frac{1}{2}$.

Step 2: Calculation

Suppose that there are $n$ people.

The probability that non of them has the same birthday as me is simply

${\left(\frac{364}{365}\right)}^n$

Step 3: Substitution

So the probability that at least one person has the same birthday as me is greater than $\frac{1}{2}$ if and only if

$1-{\left(\frac{364}{365}\right)}^n\ge \frac{1}{2}$

$\iff {\left(\frac{364}{365}\right)}^n\le \frac{1}{2}$

$\iff n\log \left(\frac{364}{365}\right)\le \log \left(\frac{1}{2}\right)$

$\iff n\ge 252.65$.

Step 4: Final answer

The probability that at least one person has the same birthday as me is greater than $\frac{1}{2}$has found to be $253$.