warm up 1)we are drawing a single card from a standard deck of 52 find the probability of...

Warm up

1)We are drawing a single card from a standard deck of 52 find the probability of P(seven/nonface card)

2)Assume that we roll two dice and a total showing is greater than 8. What is the probability that the total is odd?

3)What is the probability that we draw either a 6 or a face card?

Expected Value

© 2010 Pearson Education, Inc. All rights reserved. Section 14.4, Slide 2

14.4


• Example: The value of several items along with the probabilities that these items will be stolen over the next year are shown. Predict what the

Expected Value

(continued on next slide)

insurance company can expect to pay in claims on your policy. Is $100 a fair premium for this policy?


• Solution: We add an expected payout column to the table. For example for the laptop, $2000 × (.02) = $40.

Expected Value



• Solution: We may now compute the expected payout.

With an expected payout of $90, a $100 premium is reasonable.

Expected Value


Expected Value


• Example: How many heads we can expect when we flip four fair coins?

Expected Value and Games of Chance


• Example: How many heads we can expect when we flip four fair coins?


• Solution: There are 16 ways to flip four coins. We could use a tree to complete the table shown.

# heads expected


• Example: Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (you also keep your $1 bet); otherwise you lose the $1. What is the expected value of this bet?

•This is an experiment with two outcomes:




• Example: Suppose you bet $1 on a single number of a 38 number roulette wheel. If you win, you get $35 (you also keep your $1 bet); otherwise you lose the $1. What is the expected value of this bet?

• Solution: This is an experiment with two outcomes:


{You win (worth +$35), You lose (worth –$1)}.



The probability of winning is .

The probability of losing is

The expected value of the bet is

You can expect to lose about 5 cents for every dollar you bet.



• Example: Assume that it costs $1 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($499 considering the $1 cost). What is the expected value of this game?



State Lottery

Cost: $1

Prize: $500

$ $


The probability of winning is .

The probability of losing is

The expected value of this game is

You can expect to lose 50 cents for every ticket you buy.



• Example: Assume that it costs $1 to play a lottery. The player chooses a three-digit number between 000 and 999, inclusive, and if the number is selected, then the player wins $500 ($499 with the $1 cost). What should the price of a ticket be in order to make this game fair?

•Let x be the price of a ticket for the lottery to be fair. Then if you win, your profit will be 500 – x and if you lose, your loss will be x.




The expected value of this game is

To be fair, we must have .




We solve this equation for x.

For this game to be fair, a ticket should cost 50¢.



• Example: A test consists of multiple-choice questions with five answer choices. One point is earned for each correct answer; point is subtracted for each incorrect answer. Questions left blank neither receive nor lose points.

Other Applications of Expected Value


a) Find the expected value of randomly guessing an answer to a question. Interpret the meaning of this result for the student.b) If you can eliminate one of the choices, is it wise to guess in this situation?


• Solution (a): The probability of guessing correctly is .

The probability of guessing incorrectly is .

The expected value is

You can expect to lose points by guessing.



• Solution (b): Eliminating one answer choice then the probability of guessing correctly is .

The probability of guessing incorrectly is .

The expected value in this case is

You now neither benefit nor are penalized by guessing.



• Example: The manager of a coffee shop is deciding on how many of bagels to order for tomorrow. According to her records, for the past 10 days the demand has been as follows:

She buys bagels for $1.45 each and sells them for $1.85. Unsold bagels are discarded. Find her expected value for her profit or loss if she orders 40 bagels for tomorrow morning.




• Solution (a): We must ultimately compute

P(demand is 40) × (the profit or loss if demand is 40) + P(demand is 30) × (the profit or loss if demand is 30).

The probability that the demand is for 40 bagels is

The probability that the demand is for 30 bagels is




If demand is 40 bagels:40($1.85 – $1.45) = 40($0.40) = $16.00 profit

If demand is 30 bagels:30($0.40) = $12.00 profit on bagels sold 10($1.45) = $14.50 loss on bagels not sold

Expected profit or loss on 40 bagels is(0.40)(16) + (0.60)(–2.50) = 4.90 profit.


warm up 1)we are drawing a single card from a standard deck of 52 find the probability of...

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