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# Probability

The framework for understanding the world around us, from sports to science.

# Probability by Outcomes

The most straightforward probability calculation is "by outcomes". When there are some number of equally likely outcomes, the probability of a "successful" outcome can be calculated as $\frac{\text{# of successful outcomes}}{\text{# of total outcomes}}.$

For example, the probability of rolling an even number on a fair, six-sided die is $$\frac{3}{6} = \frac{1}{2},$$ since there are 6 equally likely outcomes (1, 2, 3, 4, 5, and 6) and 3 "successful" outcomes (2, 4, and 6).

You have signed up for an experimental drug trial, and you overheard the doctors say that there are 400 people in the test group, and 500 people in the control group.

What is the probability that you are receiving the drug (that is, what is the probability that you're in the test group)?

A fair 20-sided die is rolled. What is the probability that the result is a number divisible by 6?

In the previous two questions, the groups of "total outcomes" and "successful outcomes" were fairly straightforward.

However, sometimes they can be less obvious, and it's important to ensure that you are counting outcomes which are actually equally likely. The next few questions explore this idea.

There are 4 dormitories at your school that you could be assigned to. The three older dormitories each house the same number of students. The newest dorm houses twice the number of students as one of the older dorms.

What is the probability that you will be assigned to the newest dormitory, given that dorm assignments are given at random and all spots are filled?

You are at a charity event, and you have just purchased 1 ticket for a raffle. Prior to the raffle, you are told that there are 200 people at the event (including yourself), and about $$\frac{3}{4}$$ of the people at the event purchased tickets to the raffle. Of those people, you approximate that people were equally likely to purchase 1, 2, or 3 tickets.

Based on these estimations, what is the probability that you win the raffle?

Ashley and Allison want to go to the movies to see either Batman or Superman. They are willing to separate and go to different movies. Each of them independently flips a fair, standard coin to decide which movie they will go to.

What is the probability that they go to the same movie?

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