Question

Shared Birthdays Find the probability that of 25 randomly selected people, at least 2 share the same birthday.

Short answer

The probability that out of 25 people, at least two share the same birthday is equal to 0.569.

Step-by-step solution

Given information

Out of 25 selected people, at least two should share the same birthday.

Probability of “at least one”

The chances that at least one of the outcomes appears is complementary to the event that none of the outcomes appears.

For any given event A, it has the following notation:

$P\left(\text{A}\text{occurring}\text{at}\text{least}\text{once}\right)=1-P\left(\text{A}\text{not}\text{occurring}\right)$

Compute the probability of having unique birthdays

The probability that out of 25 people, at least two have the same birthday is equal to one minus the probability that two people have the same birthday (all unique birthdays).

Assume that the total number of days is 365 in a year.

The favorable number of outcomes for each person decreases by one as they have unique birthdays.

The probability that all 25 people have unique birthdays is computed as follows:

$$\begin{gathered} P\left(\text{all}\text{unique}\text{birthdays}\right)=\frac{365}{365}\times \frac{364}{365}\times \frac{363}{365}\times ...\times \frac{341}{365}\left(25\text{selections}\right) \\ =0.4313 \end{gathered}$$

Compute the probability that at least two have the same birthday

The probability that at least two persons have the same birthday is computed as follows:

$$\begin{gathered} P\left(\text{at}\text{least}\text{2}\text{have}\text{the}\text{same}\text{birthday}\right)=1-P\left(\text{all}\text{unique}\text{birthdays}\right) \\ =1-0.4313 \\ =0.569 \end{gathered}$$

Therefore, the probability that at least two persons have the same birthday is equal to 0.569.