THE NATUREOF PROBABILITY

Copyright © Cengage Learning. All rights reserved.

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Copyright © Cengage Learning. All rights reserved.

13.2 Mathematical Expectation

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Expected Value

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Expected Value

Choke-up toothpaste is giving away $10,000.

All you must do to have a chance to win is send a postcard with your name on it (the fine print says you do not need to buy a tube of toothpaste). Is it worthwhile to enter?

Suppose the contest receives 1 million postcards (a conservative estimate). We wish to compute the expected value (or your expectation) of winning this contest.

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Expected Value

We find the expectation for this contest by multiplying the amount to win by the probability of winning:

EXPECTATION = (AMOUNT TO WIN) (PROBABILITY OF WINNING)

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Expected Value

What does this expected value mean? It means that if you were to play this “game” a large number of times, you would expect your average winnings per game to be $0.01.

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Expected Value

Is the “Choke-up toothpaste giveaway” game fair?

If the toothpaste company charges you 1¢ to play the game, then it is fair. But how much does the postcard cost?

If you include this cost, then there is a negative expectation. We see that this is not a fair game.

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Example 1 – Find the expected value for drawing a card

Suppose that you draw a card from a deck of cards and are paid $10 if it is an ace. What is the expected value?

Solution:

EXPECTATION = $10

$0.77

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Expected Value

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Expected Value

Since we know that p1 + p2 + ··· pn = 1, we note that in many examples some of the probabilities may be 0. For Example 1, we might have said that there are two different

payoffs: $10 if you draw an ace (probability ) and $0

otherwise (probability ), so that

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Expectation with a Cost of Playing

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Expectation with a Cost of Playing

Many games charge you a fee to play. If you must pay to play, this cost of playing should be taken into consideration when you calculate the expected value.

Remember, if the expected value is 0, it is a fair game; if the expected value is positive, you should play, but if it is negative, you should not.

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Example 6 – Decide to play or not to play

Consider a game that consists of drawing a card from a deck of cards. If it is a face card, you win $20. Should you play the game if it costs $5 to play?

Solution:

You should not play this game, because it has a negative expectation.

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Expectation with a Cost of Playing

An example of the latter is a U.S. roulette game, in which your bet is placed on the table but is not collected until after the play of the game and it is determined that you lost.

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Expectation with a Cost of Playing

A U.S. roulette wheel has 38 numbered slots (1–36, 0, and 00), as shown in Figure 13.8.

Figure 13.8

U.S. roulette wheel and board

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Expectation with a Cost of Playing

Some of the more common roulette bets and payoffs are shown. If the payoff is listed as 6 to 1, you would receive $6 for each $1 bet. In addition, you would keep the $1 you originally wagered.

One play consists of having the croupier spin the wheel and a little ball in opposite directions.

As the ball slows to a stop, it lands in one of the 38 numbered slots, which are colored black, red, or green. A single number bet has a payoff of 35 to 1.

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Example 8 – Find a roulette expectation

What is the expectation for playing roulette if you bet $1 on number 5?

Solution:

The $1 you bet is collected only if you lose. Now, you can calculate the expected value:

The expected loss is about 5¢ per play.