Question

In the new IU Lottery, you pay $1 to play, and then you pick an ordered pair of numbers (repetitions allowed) from the set (0,1,2,3,4,5,6,7,8,9). Thus, (2,5) and (4,4) would be legal picks. Next, an ordered pair of numbers is randomly selected (with replacement) from this set, and if the pair selected exactly matches yours, then you get $50. If the pair does not match yours, you get nothing? What is your expected net return (the amount you receive minus the amount you paid)? A) -.75 B) +49 C) +.50 D) none of the others E) -.25 F) -.50

Answer 1

Answer: -0.50

Step-by-step explanation

x = net winnings

There are two outcomes. Either x = 50-1 = 49 or x = 0-1 = -1, which represents a win and a loss in that order. A negative x value means a loss of money.

There is exactly one way to match the ordered pair you picked, out of 10*10 = 100 ordered pairs ranging from (0,0) to (9,9) where each coordinate is chosen from the set {0,1,2,3,4,5,6,7,8,9}.

This means 1/100 = 0.01 is the probability of winning and 99/100 = 0.99 is the probability of losing.

Let's make a chart to organize this.

`\begin{array}c \cline{1-3}\text{Outcome} & \text{X} & \text{P(X)}\\\cline{1-3}\text{Win} & 49 & 0.01\\\cline{1-3}\text{Loss} & -1 & 0.99\\\cline{1-3}\end{array}`

Form a new column where we multiply each x and P(x) value.

`\begin{array}c \cline{1-4}\text{Outcome} & \text{X} & \text{P(X)} & \text{X*P(X)}\\\cline{1-4}\text{Win} & 49 & 0.01 & 0.49\\\cline{1-4}\text{Loss} & -1 & 0.99 & -0.99\\\cline{1-4}\end{array}`

Add up the values in that new column: 0.49 + (-0.99) = -0.50 is the expected value.

We expect, on average, to lose 50 cents each time we play the game.

Because the expected value is not zero, the game is considered not mathematically fair.

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Another approach:

Let's say that you really want to win the prize. The way to guarantee this would be to buy all 100 tickets ranging from (0,0) to (9,9).

That will cost you $100.

The grand prize is $50, so the net winnings is 50-100 = -50 dollars.

Buying all the tickets guarantees you land on the prize at some point. The downside is you are out $50 even when you do win the prize.

Divide that $50 loss over the number of tickets purchased.

-50/100 = -0.50 is the expected value