Random Variables
To play the Washington State Lottery's Daily Game, you pay $1 and then choose three digits (each may be 0 through 9, with repeated digits allowed). If the three digits chosen in the daily drawing match the three digits on your ticket (in the same order) you win $500; otherwise, you win nothing. (There are variations of this game, but let's stick with the basic scenario for this example.)
Let's let the variable `X` represent your net profit when playing this game one time. We call `X` a random variable because it takes on numerical values that are determined through a random process.
The possible values that `X` may take on are $499 (the $500 prize if you win, minus the $1 you paid to play the game) or -$1 (in other words, the dollar you lose when you don't win the game). Let's construct a probability model for this game:
| outcome | profit | probability |
| `x` | `P(X=x)` | |
| win | $499 | 0.001 |
| lose | -$1 | 0.999 |
A probability model (sometimes called a probability distribution) lists the values the variable might take on ($499 and -$1 in this example) along with the probability of each of those outcomes. The variable `x` here stands in for those values. The first line says that P(X=$499) = 0.001 while the second line says that P(X=-$1) = 0.999. A probability model might also include a description of each outcome ("win" and "lose" in this example).
To compute the probabilities listed above we note that there are 10 possible digits for each of the three positions on the ticket, so the probability of matching any one digit is `(1)/(10) = 0.1`. Since each digit is selected independently of the others, we can compute:
`P( mbox{win}) = P(mbox{match and match and match}) = P(mbox{match}) times P(mbox{match}) times P(mbox{match} = 0.1 times 0.1 times 0.1 = 0.001`
and then note that
`P(mbox{lose}) = 1 - P(mbox{win}) = 1 - 0.001 = 0.999`
Mean of a probability model
Imagine you really wanted to win this lottery. You could go to the store and spend $1000 to buy one of each ticket (from 000 through 999). This would ensure you of winning $500 with one of the tickets (for a $499 profit), but the other 999 would be losers (for a $999 loss). What would you average winnings be on a per-ticket basis?
`($499 - $999)/(1000) = -$500/1000 = -$0.50`
We call this the mean of the probability model. We can also rewrite the computation like this:
`($499-$999)/(1000) = $499(1/1000) - $1(999/1000) = $499 times 0.001 + (-$1) times 0.999`
Notice that this is the same result as if we multiplied each numerical outcome in the table above by its associated probability and then added the results:
`mu = E(X) \ = \ sum x cdot P(X=x) \ = \ (499)(0.001) + (-1)(0.999) \ = \ -0.50`
We use the notation E(X) here to stand for the expected value of the random variable X: the mean profit per ticket is the value we expect to get, on average, from each ticket.
For this game, each time you buy a ticket you lose an average of $0.50. Of course, you never actually lose $0.50: you either lose $1 or gain $499 (-$0.50 is just an average).
Standard deviation of a probability model
We can also compute the standard deviation of the expected winnings but (as we will always do) we start with the variance:
`Var(X) \ = \ sum (x-mu)^2 cdot P(X=x) = (499-(-0.5))^2 cdot 0.001 + (-1-(-0.5))^2 cdot 0.999 = 249.75`
This looks an awful lot like the formula for variance we saw before, but we multiply the squared deviation by the associated probability in each case instead of dividing by n−1.
Next we can take the square root of the variance to get the standard deviation:
`sigma = SD(X) = sqrt{249.75} approx $15.80`
We use μ and σ to represent the mean and standard deviation here because in a probability model we consider all possible lottery tickets, not just a small sample of certain lottery tickets.
Using the calculator
You can also compute expected value and standard deviation using the TI-84. Enter the numerical outcomes into one list (L1, say) and the associated probabilities into another list (L2, for example):
Next, press QUIT, then type 1-VarStats L1,L2 and press ENTER:
You should see the expected value (-$0.50) at the top and the standard deviation ($15.80) further down:
Exercises
1. Gambling A game involves rolling two dice: a standard six-sided die as well as and 8-sided die. If the numbers on the dice match, you win $5, otherwise you lose $1.
a) Construct a probability model for this game.
b) Compute the expected value and standard deviation for this game.
c) Should you play this game?
2. Powerball The probabilities of winning various prizes in the Powerball lottery appear below. A ticket costs $2.
numbers matched prize probability
5 + powerball $40,000,000 0.000000005707
5 $1,000,000 0.000000194038
4 + powerball $10,000 0.000001540889
4 $100 0.000052390225
3 + powerball $100 0.000081667120
3 $7 0.002776697951
2 + powerball $7 0.001415568421
1 + powerball $4 0.009024456277
0 + powerball $4 0.018047283884
a) What outcome is missing from the table?
b) Compute the probability for this missing outcome.
c) Compute the expected value of your prize money when you buy one ticket.
d) Compute the expected value of the profit when you buy one ticket.
e) Should you buy a ticket for the Powerball? Explain.
f) The top prize is guaranteed to be a minimum of $40,000,000, but rises each time no one wins the jackpot. The largest such jackpot was $365 million, for the drawing on February 18, 2006. Compute the expected value of your profit when buying one Powerball ticket if the top prize is $365,000,000 instead of $40,000,000.
g) Should you buy a ticket if the jackpot is $365,000,000? Explain.