1.
First, what should we mean by probability? If you roll a fair six-sided die, what is the probability of rolling a 6?
What is the probability of rolling an even number?
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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?
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.
2.
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.
(a)
Suppose you roll two dice (one red and one green). How many total outcomes are there?
(b)
Of those outcomes, how many have different numbers on the two dice?
Hint.
How many sequences of two different numbers can you make using the numbers 1 to 6?
(c)
Combining the two numbers your found above, what is the probability that two dice will show different numbers?
(d)
What is the probability that you will get three different numbers when rolling three dice? (Assume the dice are different colors).
3.
Now to birthdays. There are 365 days in a year.
(a)
How many possible sequences of 30 birthdays are there?
(b)
How many possible sequences of 30 birthdays contain no repeats?
(c)
What is the probability that 30 people have no repeated birthdays?
(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?
(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?
