Expected Value: Day 2
Today is an opportunity to go into greater depth with modeling and the concept of expected value. Spreadsheets such as Excel might be well-used (but are not necessary) for today's activities.
The students should be divided into groups to work on the following problems (encourage them to devise models first and try to find trends that might lead to more analytical solutions):
- If you toss a coin 10 times:
- What is the most number of heads you can get?
- What is the least number of heads you can get?
- About how many heads do you expect to get?
- If you toss a coin 100 times:
- What is the most number of heads you can get?
- What is the least number of heads you can get?
- About how many heads do you expect to get?
- If you roll a die 10 times and add up the numbers from each roll:
- What is the highest total you can get?
- What is the lowest total you can get?
- About what do you expect the total to be?
- If you roll a die 100 times and add up the numbers from each roll:
- What is the highest total you can get?
- What is the lowest total you can get?
- About what do you expect the total to be?
From these activities there should be a bridge built from probability to expected value. Once a definition for expected value has been discovered through experimentation and class discussion, go on to the final problem:
There are 3 boxes, one of which has $1 in it and the other two have nothing in them. It costs $1 to play the game. If you choose the right box, you get your $1 back plus the $1 from the box. If you choose the wrong box, you lose your $1.
- How many games out of 10 do you expect to win?
- How many games out of 100 do you expect to win?
- How much money would you expect to win (or lose) if you played 10 games?
- How much money would you expect to win (or lose) if you played 100 games?
- What is the expected value of your winnings (or losings) per game?
If you like, you may return to the introduction page.
Go on to the lesson plan for Day 3