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 [latex]\frac{1}{38}[/latex]. The complement, the probability of losing, is [latex]\frac{37}{38}[/latex].

Summarizing these along with the values, we get this table:

Notice that if we multiply each outcome by its corresponding probability we get [latex]\$35\cdot \frac{1}{38}=0.9211[/latex] and [latex]-\$1\cdot \frac{37}{38}=-0.9737[/latex], and if we add these numbers we get

0.9211 + (-0.9737) ≈ -0.053, which is the expected value we computed above.

Earlier, we calculated the probability of matching all 6 numbers and the probability of matching 5 numbers:

[latex]\frac{{}_{6}{{C}_{6}}}{{}_{48}{{C}_{6}}}=\frac{1}{12271512}\approx0.0000000815[/latex] for all 6 numbers,

[latex]\frac{\left({}_{6}{{C}_{5}}\right)\left({}_{42}{{C}_{1}}\right)}{{}_{48}{{C}_{6}}}=\frac{252}{12271512}\approx0.0000205[/latex] for 5 numbers.

Our outcome values and the probability of each outcome are shown in the table below.

The expected value, then is:

[latex]\left(\$999,999 \right)\cdot \frac{1}{12271512}+\left( \$999\right)\cdot\frac{252}{12271512}+\left(-\$1\right)\cdot\frac{12271259}{12271512}\approx-\$0.898[/latex]

On average, one can expect to lose about 90 cents on a lottery ticket. Of course, most players will lose $1.

The probabilities and outcomes are

The expected value is ($99,725)(0.00242) + (-$275)(0.99758) = -$33.

Not surprisingly, the expected value is negative: the insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people.

For people buying the insurance, there is a negative expected value, but there is a security that comes from insurance that is worth that cost.