Take another guess: A student takes a multiple-choice test that has 7 questions. Each question has five choices. The student guesses randomly at each answer. Round the answers to three decimal places.
Part 1 of 2
(a) Find $P(4)$.
\[
P(4)=
\]

Part 2 of 2
(b) Find $P$ (More than 2 ).
\[
P(\text { More than } 2)=
\]

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Step 1 ：Calculate the combinations for 7 items taken 4 at a time: \(C(7, 4) = \frac{7!}{4!(7-4)!} = 35\)

Step 2 ：Calculate the probability of guessing 4 questions correctly: \((1/5)^4 = 0.0016\)

Step 3 ：Calculate the probability of guessing the remaining 3 questions incorrectly: \((4/5)^3 = 0.512\)

Step 4 ：Multiply these values together to find the probability of guessing exactly 4 questions correctly: \(P(4) = 35 * 0.0016 * 0.512 = 0.028672\)

Step 5 ：\(\boxed{P(4) \approx 0.029}\) when rounded to three decimal places

Step 6 ：Calculate the combinations and probabilities for guessing exactly 3, 5, 6, and 7 questions correctly in a similar way

Step 7 ：\(P(3) = C(7, 3) * (1/5)^3 * (4/5)^4 = 0.103\)

Step 8 ：\(P(5) = C(7, 5) * (1/5)^5 * (4/5)^2 = 0.006\)

Step 9 ：\(P(6) = C(7, 6) * (1/5)^6 * (4/5)^1 = 0.001\)

Step 10 ：\(P(7) = C(7, 7) * (1/5)^7 * (4/5)^0 = 0.000\)

Step 11 ：Add these probabilities together to find the probability of guessing more than 2 questions correctly: \(P(\text { More than } 2) = 0.103 + 0.029 + 0.006 + 0.001 + 0.000 = 0.139\)

Step 12 ：\(\boxed{P(\text { More than } 2) \approx 0.139}\) when rounded to three decimal places