Answer:
the probability that Eric gets fewer than 5 questions correct is approximately 0.857, or 85.7%.
Explanation:
Let X be the number of questions Eric gets correct. Since there are 7 questions and each has 4 choices, the probability of guessing any one question correctly is 1/4, and the probability of guessing any one question incorrectly is 3/4. Since Eric is guessing on every question, we can model X as a binomial random variable with n = 7 and p = 1/4.
To find the probability that Eric gets fewer than 5 questions correct, we need to calculate:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Using the binomial probability mass function, we get:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where (n choose k) is the binomial coefficient, which gives the number of ways to choose k items from a set of n items.
Plugging in n = 7 and p = 1/4, we get:
P(X = k) = (7 choose k) * (1/4)^k * (3/4)^(7 - k)
Therefore, we have:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
= (7 choose 0) * (1/4)^0 * (3/4)^7 + (7 choose 1) * (1/4)^1 * (3/4)^6
+ (7 choose 2) * (1/4)^2 * (3/4)^5 + (7 choose 3) * (1/4)^3 * (3/4)^4
+ (7 choose 4) * (1/4)^4 * (3/4)^3
≈ 0.857
Therefore, the probability that Eric gets fewer than 5 questions correct is approximately 0.857, or 85.7%.