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# Why Doesn't X + X = 2X? This blog post comes from AP Stats guru Daren Starnes. Daren has taught AP Statistics since the course launched in 1996-97. He has served as a reader, table leader, and question leader for the AP Statistics exam for 20 years. Daren has led over 200 workshops for statistics teachers, both online and in-person. He is also lead author of two popular statistics textbooks.

In Algebra, x + x = 2x. Does a similar result hold in Statistics? That is, if X is a random variable, does X + X = 2X? Let’s investigate with a simple game of roulette.

An American roulette wheel has 38 slots numbered 1 through 36, plus 0 and 00. Half of the slots from 1 to 36 are red; the other half are black. Both the 0 and 00 slots are green. Suppose that a player places a \$1 bet on red. If the ball lands in a red slot, the player gets the original dollar back, plus an extra dollar for winning the bet. If the ball lands in a different-colored slot, the player loses the \$1 bet. Let X = the player’s profit on a single \$1 bet on red. Because there is an 18/38 chance that the ball lands in a red slot, the probability distribution of X is as shown in the table. The random variable X has mean Interpretation: The player is expected to lose about 5 cents, on average, when placing a \$1 bet on red in a game of roulette. (And the casino expects to win about 5 cents per game from such a player.)

The standard deviation of X is Interpretation: The player’s profit when placing a \$1 bet on red in a game of roulette typically varies by about \$1 from the expected loss of 5 cents.

## Combining Random Variables: X + X

Suppose a gambler plays two games of roulette, each time placing a \$1 bet on red. What can we say about the player’s total profit in two games, X + X?

• If the player wins both games, the total profit is \$1 + \$1 = \$2. The probability that happens is (18/38)(18/38) = 0.2244, since the outcomes of individual games are independent.

• If the player loses both games, the total profit is −\$1 + −\$1 = −\$2. The probability that happens is (20/38)(20/38) = 0.2770.

• If the player wins one game and loses the other, the total profit is \$1 + −\$1 = \$0. There are two ways this can happen—win then lose or lose then win—so the probability is (18/38)(20/38) + (20/38)(18/38) = 0.4986.

Here is the probability distribution of X + X. The player’s expected profit in two games is The standard deviation of X + X is Interpretation: The player’s total profit when placing a \$1 bet on red in two games of roulette typically varies by about \$1.41 from the expected loss of 10.5 cents.

Note that the mean of X + X can also be found as follows: 