Hey there, let's walk through this multiple-choice exam score analysis—this is a classic probability problem, so let's break it down into digestible parts.

First, let's calculate the probability you get any individual question right. Here's the breakdown:

• You know 60% of the material, so for those questions, you're guaranteed a correct answer (probability = 0.6).
• For the remaining 40% of questions you don't know, you guess randomly from 4 options. The chance of guessing correctly is 1/4 = 0.25.

Combining these two scenarios, your overall single-question success rate is:
P(correct) = 0.6 + (0.4 * 0.25) = 0.7

That means you have a 70% chance to get any given question right.

The expected score tells you the average score you'd get if you took this exam many, many times. Since each question is independent:

• The expected score for one question is equal to its success probability (1 point for correct, 0 for incorrect).
• Multiply that by the total number of questions n to get the total expected score:

E[Total Score] = n * 0.7

For example, if there are 100 questions, your expected score is 70.

Variance measures how much your actual score might deviate from the expected value. For a single question (a Bernoulli trial), variance is P*(1-P). Since all questions are independent, we multiply by n for the total variance:

Var(Total Score) = n * 0.7 * 0.3 = 0.21n

The standard deviation (the square root of variance) gives a more intuitive sense of spread:
SD(Total Score) = sqrt(0.21n)

Using the 100-question example again:

• Variance = 21
• Standard deviation ≈ 4.58

This means you'd expect most of your scores to fall within ~±4.6 points of the expected 70 (so between 65 and 75).

• Small n (e.g., n < 30): Your total score follows a binomial distribution Binomial(n, 0.7). You can calculate exact probabilities for specific scores using the binomial probability formula:
  P(X = k) = C(n, k) * (0.7)^k * (0.3)^(n-k)
  where C(n, k) is the combination of n items taken k at a time.
• Large n (e.g., n ≥ 30): Thanks to the Central Limit Theorem, your total score will approximate a normal distribution with mean 0.7n and variance 0.21n. This lets you use normal distribution tables or calculators to estimate probabilities like "what's the chance I score above 80%?"

Suppose n = 50 questions, and you want to know the probability of scoring at least 40 points:

• Expected score = 50 * 0.7 = 35
• Variance = 50 * 0.21 = 10.5, standard deviation ≈ 3.24
• Calculate the z-score: z = (40 - 35)/3.24 ≈ 1.54
• Looking up z=1.54 in a standard normal table, the probability of scoring 40 or higher is ~6.18%.

内容的提问来源于stack exchange，提问作者White Mamba