Expected Value and Fair Game S-MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S-MD.7 (+) Analyze.

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Presentation transcript:

Expected Value and Fair Game S-MD.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). S-MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Expected Value  The sum of the products of each value of each outcome and the corresponding probability of the outcome  Steps:  Find the probability of each event  Multiply the value of each event by the probability of the event  Add all results together

Probability Review  Basic probability: Divide the number of desired outcomes by the total number of outcomes EX: Find the probability of choosing a card that is not a seven from a standard deck of cards.  And probability: Multiply the probabilities EX: Find the probability of choosing a red card and a face card.  Or probability: Add the probabilities EX: Find the probability of choosing a card that is a jack or a king.

Example 1  You participate in a game show where you have to answer questions that have four possible answers. You get $100 for every answer that you get right and lose $60 for every answer that you get wrong. Find the expected value for each question to find out if you should guess.

Example 2  In a game you flip a coin twice, and record the number of heads that occur. You get 10 points for 2 heads, zero 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?

Fair Game  Fair game: the cost to play the game is less than or equal to the expected value  EX: A spinner has four regions of equal area number 1-4. If you land on section 1 or section 3 you get $0. If you land on section 2 you get $25 and if you land on section 4 you get $15. What should the cost of the game be in order for it to be fair?

Example 3  In a carnival game, players win prizes by rolling a cube. The cube has one red side, one white side, one blue side, and three green sides. The game costs $1. If the cube lands on the red side, the player gets $0.50. If it lands on the white side, the player wins $1. If the cube lands on the blue side, the player wins $1.50. Green sides are worth nothing.  Find the expected value.  Should you play this game?