Warm Up

If Babe Ruth has a 57% chance of hitting a home run every time he is at bat, run a simulation to find out his chances of hitting a homerun at least 3 times if he gets to bat 4 times in a game. Run 10 trials.

Homework Check – pg 15, #s 2 & 3

2) 1 – 71 represent a made free throw 72-100 are misses Run RandInt(1, 100, 2) 3) 1 - 20= blue 21 - 50= green 51 - 90= red 91 - 100 = multi-colored ballRandInt (1, 100, 4)

Day 8: Expected Value

Unit 1: Statistics

Today’s Objectives

• Students will work with expected value.

Expected Value and Fair Games

Expected Value

Expected value is the weighted average of all possible outcomes.

For example, if a game has the outcomes of winning $10, $20 and $60.

The average of $10, $20, and $60 = $30

This assumes an even distribution: meaning each outcome ($10, $20 or $30) has the same probability of occurring.

10 20

60

Sometimes, outcomes will not have equal likelihoods (probabilities). For example in this spinner which are you more likely to land on?

What is the probability you will get a 1?

What is the probability you will get a 2?

What is the probability you will get a 3?

Example 1Consider a die-rolling game that costs $10

per play. A 6-sided die is rolled once, and your cash winnings depend on the number rolled. Rolling a 6 wins you $30; rolling a 5 wins you $20; rolling any other number results in no payout.

Outcomes

Probability

Value

Total

Therefore, when playing the dice game, you should expect to lose $1.67 per game played.

In the example above, it was determined that the expected winnings of the game were - $1.67 per roll. This is an impossible outcome for one game; you can only either lose $10, win $10, or win $20. However, the expected value is useful as a long-term average figure. If you play this dice game over and over, you will lose somewhere near $1.67 per game on average.

• Another way to think of expected value is assigning the game a particular cost (or benefit) of playing; you should only decide to play the game if the fun of playing is worth paying $1.67 each time.

• The more times the situation is repeated, the more accurately the expected value will mirror the actual average outcome. For example, you might play the game five times in a row and lose every time, resulting in an average loss of $10. However, if you were to play the game 1,000 times or more, your average result would almost always be close to the expected value of -$1.67 per game.

• This principle is called the "law of large numbers."

Example 2 (pg 19 #6)A $20 bill, two $10 bills, three $5

bills and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen?

Outcomes

Probability

Value

Total

Example 3 – pg 20 #7In a game, you flip a coin twice, and record the

number of heads that occur. You get 10 points for 2 heads, 0 points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn?

You play a game in which you roll one fair die. If you roll a 6, you win $5. If you roll a 1 or a 2, you win $2. If you roll anything else, you don’t win any money.

You try!

At Tucson Raceway Park, your horse, My Little Pony, has a probability of 1/20 of coming in first place, a probability of 1/10 of coming in second place, and a probability of ¼ of coming in third place. First place pays $4,500 to the winner, second place $3,500 and third place $1,500. Is it worthwhile to enter the race if it costs $1,000?

What does an expected value of -$50 mean?

Its important to note that nobody will actually lose $50—this is not one of the options. Over a large number of trials, this will be the average loss experienced.

This is the Law of Large Numbers!

Insurance companies and casinos build their businesses based on the law of large numbers.

Questions about expected value?

Anyquestions about statistics?

Homework

• Complete the handout