Transcript Day 8: Expected Value

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 ball
RandInt (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 1
Consider 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 #7
In 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 try!
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.
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?
Any
questions
about
statistics?
Homework
• Complete the handout