随机变量场景下柠檬水摊盈利概率求解的正确性验证

Hey there! Nice job setting up the profit equation and framing the problem with random variables—you’ve got the core logic right for translating between profit (X) and number of drinks sold (Y). Let’s walk through your solution and catch a key detail about the distribution of (Y) that’s important here:

1. Calculating (F_X(27.38)) #

First, your profit formula is spot-on:
$$X = 21Y - 20(64 - Y) = 41Y - 1280$$
And converting the inequality (X ≤ 27.38) to (Y ≤ 31.88) (so (Y ≤ 31) since (Y) is an integer number of drinks) is totally correct.

Here’s the key correction: The problem states (Y) (number of purchases) has equally likely values from 0 to 64. That means (Y) follows a discrete uniform distribution, where each outcome (Y=i) has probability (\frac{1}{65}) (since there are 65 total possible values: 0 through 64). You incorrectly used a binomial distribution here—binomial would apply if each drink had a 50% chance of being sold independently, but that’s not what the problem says.

So the correct calculation for (P(X ≤ 27.38) = P(Y ≤ 31)) is the number of valid (Y) values divided by total possible values:
$$P(Y ≤ 31) = \frac{32}{65} ≈ 0.4923$$
(That’s 32 values: 0,1,...,31)

2. Probability of no loss ((X ≥ 0)) #

Your inequality conversion is again correct:
$$0 ≤ 41Y - 1280 \implies Y ≥ \frac{1280}{41} ≈ 31.21$$
So (Y ≥ 32) (since (Y) must be an integer).

Using the correct uniform distribution, the number of valid (Y) values is (64 - 32 + 1 = 33). So the probability is:
$$P(Y ≥ 32) = \frac{33}{65} ≈ 0.5077$$

Quick Recap of Your Wins & Fixes #

• ✅ Profit equation (X = 41Y - 1280) is perfect.
• ✅ Inequality transformations to solve for (Y) are logically sound.
• ❌ Fixed: Misapplied binomial distribution instead of discrete uniform (since all (Y) values are equally likely, not each drink having a 50% sell chance).

Hope that clears things up—great work getting the foundational pieces right!

备注：内容来源于stack exchange，提问作者elephanty