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Using Probability to Make Fair Decisions

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Did you know that you can use probability to help you make a fair decision? Knowing how likely an event is to happen can help you determine whether or not to accept a task. Learn more in this lesson.

Probability

What is probability? Probability refers to how likely something is to happen. Many things in the real world have probability attached to them. For example, the chance of rain each day is a probability (60 percent chance of rain). Your chances of winning a game, the lottery, or anything else is also a probability such as 1:1,000,000. The smaller the probability, the less likely the specified event will happen.

It is probability theory that gives you your probability numbers and helps you to better understand what is going on so you can make fair decisions, such as whether you should bring an umbrella with you tomorrow or not.

Probability theory all started with the problem of points a long time ago.

Problem of Points

The problem was first posed by Chevalier de Mere in 1654 to the mathematician Blaise Pascal. Pascal then continued discussing the problem and its solution with another mathematician, Pierre de Fermat.

Let's take a look at this famous problem from years ago.

Imagine this scenario. Two equally skilled players are deep in the middle of a flip-the-coin game when one player receives an urgent phone call. This player must stop playing and leave right away. This player though has a lead of 9 to 8. The players have each put in $50 into the game, so the winner gets to take home $100. The first to 10 wins the game. The problem of points arises here with this question: Since the game has been cut short, how should the $100 winnings be split between the players? For it to be a fair split, it must represent the likelihood of the player with the lead winning vs. the likelihood of the other player winning.

Making a Fair Decision

What did Pascal propose in his conversations with Fermat?

Pascal pointed out that in order to make a fair determination, he had to assess all possible outcomes from the time the player in the lead had to go. From those possible outcomes, he can then determine how many of those outcomes result in the player with the lead winning. To determine the probability of the player with the lead actually winning, Pascal took the number of winning outcomes and divided it by the total number of possible outcomes. Then, to find the fair amount of the winnings to give to the player with the lead, Pascal multiplied the probability with the $100 winnings. The other player would get the rest.

For the flip-the-coin game, Pascal figured that only 2 more rounds were needed for one of the players to get to 10 since the player with 9 points only needed 1 more win and the other player 2 more wins. Listing all the possible outcomes, he gets this:

h h
h t
t h
t t

There are four possible outcomes to finish the game. The player with 9 points wins if he gets one more heads. Out of these four possible outcomes, three of them result in the player with 9 points winning. Dividing the 3 winning outcomes by the 4 total possible outcomes, the probability of winning is 3:4 or 75 percent. Multiplying this 75 percent with the $100 winnings, the fair amount to give to the player with 9 points is $100 * 0.75 = $75. The other player gets the rest: $100 - $75 = $25.

There are other ways of making this calculation such as my using combinations or even Pascal's Triangle. But these won't be discussed in this lesson.

Example

Let's look at another example.

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