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what's the probability that in a classroom of 30 students, at least two people s...

Feb 15, 2024

what's the probability that in a classroom of 30 students, at least two people share the same birthday?

Solution by Steps

step 1

To find the probability that at least two people in a classroom of 30 students share the same birthday, we use the complement of the probability that no one shares a birthday

step 2

The probability that no two people share a birthday in a group of 30 can be calculated by multiplying the probabilities that each subsequent person has a birthday different from those already considered

step 3

The probability for the first person is 365/365, for the second person it is 364/365, for the third person 363/365, and so on until the 30th person

step 4

The probability that no one shares a birthday is

\frac{365}{365} \times \frac{364}{365} \times \frac{363}{365} \times \cdots \times \frac{336}{365}

step 5

The probability that at least two people share a birthday is 1 minus the probability that no one shares a birthday

step 6

Using the asksia-ll calculator, the probability that at least two people share a birthday in a classroom of 30 is approximately 0.7063

Answer

The probability that at least two people in a classroom of 30 students share the same birthday is approximately 0.7063 or ≈ 1 in 1.4.

Key Concept

Complementary Probability

Explanation

To calculate the probability of an event happening, we can find the probability of the event not happening and subtract it from 1. This is known as the complementary probability. In the birthday problem, it's easier to calculate the probability that no one shares a birthday and then subtract that from 1 to find the probability that at least two people share a birthday.