Question

To determine the smallest number of people you need to choose at random so that the probability that at least two of them were both born on April $\mathbf{1}$exceeds$\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$.

Short answer

Answer

The smallest number of people is $613$.

Step-by-step solution

Step 1. Given information

Two of them were both born on April $1$exceeds $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

Step 2. Definition and formula to be used

Probability deals with the occurrence of a random event.

Step 3. Compute probability

The probability that someone is born on April first is$\frac{1}{365}$

The probability that someone isn’t born on April first is$\frac{364}{365}$

The probability that no one out of$n$ people is born on April first is$n\times \frac{1}{365}{\left(\frac{364}{365}\right)}^{n-1}$

The probability that at least $2$out of $n$people is born on April first is

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

We just compute when this is greater than $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$and we get $n=613$.

The smallest number of people is $613$.