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Suppose you were in a class of 30 students. How likely is it that at least two of the students were born on the same day of the year?

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Assume that all days are equally likely and that nobody was born on February 29th. Would you believe the answer is more than 25%? More than 50%? More than 70%??? Let’s find the answer.

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1. Probability die.

First, what should we mean by probability? If you roll a fair six-sided die, what is the probability of rolling a 6?

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What is the probability of rolling an even number?

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2. Multiple dice.

We will define the probability of an event as the number of ways the event can happen divided by the total number of things that can happen.

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(a)

Suppose you roll two dice (one red and one green). How many total outcomes are there?

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(b)

Of those outcomes, how many have different numbers on the two dice?

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Hint.

How many sequences of two different numbers can you make using the numbers 1 to 6?

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(c)

Combining the two numbers your found above, what is the probability that two dice will show different numbers?

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(d)

What is the probability that you will get three different numbers when rolling three dice? (Assume the dice are different colors).

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3. Birthdays.

Now to birthdays. There are 365 days in a year.

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(a)

How many possible sequences of 30 birthdays are there?

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(b)

How many possible sequences of 30 birthdays contain no repeats?

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(c)

What is the probability that 30 people have no repeated birthdays?

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(d)

Among the 30 people, either they all have different birthdays or at least two share a birthday. Since this is certain, its probability is 1. So what is the probability that at least two people (out of the 30) share a birthday?

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(e)

What is the smallest number of people you would need to have a greater than 90% chance that at least two share a birthday?

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