4.2 Mean or Expected Value and Standard Deviation

Expected value as the long-run average of repeated trials, the standard deviation of a distribution table, and deciding whether a wager is fair.

Statistics, 4.2 Mean or Expected Value and Standard Deviation

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22 exercises

4.2 Exercise 1

A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value?

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4.2 Exercise 2

You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. You pay $1 to play. If you guess the right suit every time, you get your money back and $256. What is your expected profit of playing the game over the long term?

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4.2 Exercise 3

Suppose you play a game with a biased coin. You play each game by tossing the coin once. P(heads) = $\frac{2}{3}$ and P(tails) = $\frac{1}{3}$ . If you toss a head, you pay $6. If you toss a tail, you win $10. If you play this game many times, will you come out ahead?

a. Define a random variable X.

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4.2 Exercise 4

b. Complete the following expected value table.

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4.2 Exercise 5

c. What is the expected value, μ? Do you come out ahead?

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4.2 Exercise 6

Suppose you play a game with a spinner. You play each game by spinning the spinner once. P(red) = $\frac{2}{5}$ , P(blue) = $\frac{2}{5}$ , and P(green) = $\frac{1}{5}$ . If you land on red, you pay $10. If you land on blue, you don't pay or win anything. If you land on green, you win $10. Complete the following expected value table.

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4.2 Exercise 7

Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table like the table and calculate the mean μ and standard deviation σ of X.

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4.2 Exercise 9

Find the expected value from the expected value table.

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4.2 Exercise 11

Identify the mistake in the probability distribution table.

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4.2 Exercise 13

Define the random variable X.

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4.2 Exercise 15

Find the probability that a physics major will do postgraduate research for four years. P(x = 4) = ________

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4.2 Exercise 17

On average, how many years would you expect a physics major to spend doing postgraduate research?

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4.2 Exercise 19

In words, define the random variable X.

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4.2 Exercise 21

P(x < 4) = ________

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4.2 Exercise 23

What does the column P(x) sum to and why?

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4.2 Exercise 25

You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. What is the expected value of playing the game?

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4.2 Exercise 28

A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

If the card is a face card, and the coin lands on heads, you win $6.
If the card is a face card, and the coin lands on tails, you win $2.
If the card is not a face card, you lose $2, no matter what the coin shows.

Find the expected value for thi…

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4.2 Exercise 30

Complete the PDF and answer the questions.

Find the probability that x = 2.
Find the expected value.

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4.2 Exercise 32

A venture capitalist, willing to invest $1,000,000, has three investments to choose from: The first investment, a software company, has a 10 percent chance of returning $5,000,000 profit, a 30 percent chance of returning $1,000,000 profit, and a 60 percent chance of losing the million dollars. The second company, a hardware company, has a 20 percent chance of returning $3,000,000 profit, a 40 perc…

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4.2 Exercise 34

Suppose that the PDF for the number of years it takes to earn a bachelor of science (B.S.) degree is given as in the table.

On average, how many years do you expect it to take for an individual to earn a B.S.?

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4.2 Exercise 36

A “friend” offers you the following deal: For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift.

Ten of the coupons are for a free gift worth $6.
Eighty of the coupons are for a free gift worth $8.
Six of the coupons are for a free gift worth $12.
Four of the coupons are for a free gift worth $40.

Ba…

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4.2 Exercise 38

In a raffle, there are 250 prizes of $5, 50 prizes of $25, and 10 prizes of $100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to charge to break even?

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