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Short-year birthday collision

ProbabilityMedium

On a small planet a year has $10$ days. $3$ people have birthdays that are independent and uniform over these days. What is the probability that at least $2$ of them share a birthday?

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Hint

Compute the probability that all birthdays differ and subtract it from $1$ .

Solution

Count equally likely outcomes: $1 0^{3} = 1000$ .
Outcomes with all $3$ birthdays different: $10 \times 9 \times 8 = 720$ .
The probability that all differ is $\frac{720}{1000}$ .
So the probability of at least one shared birthday is $1 - \frac{720}{1000} = \frac{280}{1000}$ .
In lowest terms this is $\frac{7}{25}$ .

Answer: $\frac{7}{25}$

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