The birthday problem involves calculating the probability that at least two people in a group share the same birthday. There are several ways to solve it, but one of the most common approaches is to use complementary probabilities. In other words, we can calculate the probability that no two people share the same birthday, and then subtract it from 1 to obtain the probability of at least one shared birthday.
Let’s denote the total number of people in the room as n, which in this case is 23. We’ll also assume there are 365 possible birthdays (ignoring leap years). For simplicity, we will assume that each person’s birthday is equally likely and independent of the others’.
Let’s denote the probability of no shared birthdays as P(no shared birthdays). To calculate this, we will first compute the probability that the second person has a different birthday than the first person, the third person has a different birthday than the first two, and so on.
The first person has a birthday, and there are no restrictions on it. So out of 365 days, there are 365 possibilities for the first person’s birthday. The probability that the first person has a birthday is therefore:
$$\frac{365}{365} = 1$$
The second person also has a birthday, and we want it to be different from the first person. There are now 364 remaining days, so the probability of this is:
$$\frac{364}{365}$$
For the third person, we want their birthday to be different from the first two, which leaves 363 possible days:
$$\frac{363}{365}$$
We can continue this process, computing the probability that the 23rd person has a different birthday:
$$\frac{365 - (23 - 1)}{365} = \frac{343}{365}$$
Now we multiply all these probabilities together to get the overall probability of no shared birthdays:
$$P(\text{no shared birthdays}) = \frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{343}{365}$$
To calculate the probability of at least one shared birthday, we can subtract the probability of no shared birthdays from 1:
$$\begin{aligned} P(\text{at least one shared birthday}) &= 1 - P(\text{no shared birthdays}) \\ &= 1 - \left(\frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{343}{365}\right) \end{aligned}$$
Using a calculator to compute this expression, we get:
P(at least one shared birthday) ≈ 1 − 0.4927 = 0.5073
There is approximately a 50.73% chance that at least two of the 23 people in the room share the same birthday.