概率（期望）问题：计算10张优惠券中不同类型数量的期望

安全阿岳

2026-4-21

Hey there! This is a classic expectation problem where using indicator random variables makes the solution way simpler than trying to compute the expectation directly. Let me walk you through each step clearly:

Step 1: Define Indicator Variables

Instead of counting the total distinct types upfront, let's create 20 indicator variables (one for each coupon type):

Let $X_i = 1$ if the $i$-th coupon type appears at least once in our 10-coupon set.
Let $X_i = 0$ if the $i$-th coupon type doesn't appear at all.

The total number of distinct coupon types we end up with (let's call this $X$) is just the sum of all these indicators:
$X = X_1 + X_2 + \dots + X_{20}$

Step 2: Use Linearity of Expectation

Linearity of expectation is super powerful here—it works even when variables aren't independent! The expected value of $X$ equals the sum of the expected values of each $X_i$:
$E[X] = E[X_1] + E[X_2] + \dots + E[X_{20}]$

Step 3: Calculate $E[X_i]$

For any single coupon type $i$, $E[X_i]$ is the probability that this type appears at least once in the 10 coupons. It's easier to calculate the complement first: the chance that none of the 10 coupons are type $i$.

Each coupon has a $\frac{19}{20}$ chance of not being type $i$. For 10 independent coupons, this probability is $\left( \frac{19}{20} \right)^{10}$.

So the probability that type $i$ does appear is:
$E[X_i] = 1 - \left( \frac{19}{20} \right)^{10}$

Step 4: Compute the Total Expectation

Since all 20 coupon types have identical probabilities, each $E[X_i]$ is the same. Multiply by 20 to get the total expected value:
$E[X] = 20 \times \left( 1 - \left( \frac{19}{20} \right)^{10} \right)$