4.2 Mean or Expected Value and Standard Deviation

A customer orders 1, 2, or 3 bottles of water. The probability of ordering 1 bottle is 0.6, the probability of ordering 2 bottles is 0.3, and the probability of ordering 3 bottles is 0.1. Find the long-term average or expected value, μ, of the number of bottles a customer will order.

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.

b. Complete the following expected value table.

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

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.

Two stacks of cards are placed on a table. Each stack has cards number 1 to 8. One card from each stack is picked up. Let X = number of cards showing an odd number. Construct a table like Table 4.11 and calculate the mean µ and standard deviation σ of X.

On May 11, 2013 at 9:30 PM, the probability that moderate seismic activity (one moderate earthquake) would occur in the next 48 hours in Iran was about 21.42%. Suppose you make a bet that a moderate earthquake will occur in Iran during this period. If you win the bet, you win $50. If you lose the bet, you pay $20. Let X = the amount of profit from a bet.

P(win) = P(one moderate earthquake will occu…

Find the expected value from the expected value table.

Identify the mistake in the probability distribution table.

Define the random variable X.

Find the probability that a physics major will do post-graduate research for four years. P(x = 4) = _______

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

In words, define the random variable X.

P(x < 4) = _______

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

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?

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.

1. Find the expected value for thi…

Complete the PDF and answer the questions.

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

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

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.?

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.

…

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