In: Statistics and Probability
If you bet $1 on a specific number on roulette and win you receive $36. What is the expected value of this bet? Which is the better bet, betting on “odds” or betting on a specific number?
Agamecost$10toplay. Adieisrolled. Ifa6appearsyouwin$20. Ifa4or5appears you win $13. If a 3 appears you win $10. Otherwise you win $0. What is the expected value of the game?
A homeowner pays $155 to insure her $150000 house against alien death rays. The insurance company estimates there is a 0.1% chance of her house being destroyed by an alien death ray. What is the expected value of the policy to the insurance company?
Expected Value
Expected Value is the average gain or loss of an event if the procedure is repeated many times.
We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.
Suppose you bet $1 on each of the 38 spaces on the wheel, for a total of $38 bet. When the winning number is spun, you are paid $36 on that number. While you won on that one number, overall you’ve lost $2. On a per-space basis, you have “won” -$2/$38 ≈ -$0.053. In other words, on average you lose 5.3 cents per space you bet on.
We call this average gain or loss the expected value of playing roulette. Notice that no one ever loses exactly 5.3 cents: most people (in fact, about 37 out of every 38) lose $1 and a very few people (about 1 person out of every 38) gain $35 (the $36 they win minus the $1 they spent to play the game).
There is another way to compute expected value without imagining what would happen if we play every possible space. There are 38 possible outcomes when the wheel spins, so the probability of winning is \displaystyle \frac{1}{38}381. The complement, the probability of losing, is \displaystyle \frac{37}{38}3837.
Summarizing these along with the values, we get this table:
Outcome | Probability of outcome |
$35 | \displaystyle \frac{1}{38}381 |
-$1 | \displaystyle \frac{37}{38}3837 |
Notice that if we multiply each outcome by its corresponding probability we get \displaystyle \$35\cdot \frac{1}{38}=0.9211$35⋅381=0.9211 and \displaystyle -\$1\cdot \frac{37}{38}=-0.9737−$1⋅3837=−0.9737, and if we add these numbers we get
0.9211 + (-0.9737) ≈ -0.053, which is the expected value we computed above.