A student takes a 10 question multiple choice exam and guesses on each question Each question has five choices What is the probability of getting at least 6 correct out of the ten questions?

Answer 1

To calculate the probability of getting at least 6 correct out of 10 questions when guessing on each question, we can use the binomial probability formula. The probability of guessing a question correctly is 1/5, and the probability of guessing incorrectly is 4/5. We need to find the sum of the probabilities of getting 6, 7, 8, 9, or 10 questions correct. This involves calculating the individual probabilities of each scenario using the binomial probability formula and then adding them together.

Answer 2

By guessing, the probability of getting at least 6 correct is 62201/9765625 which is approx 0.0064 or 0.64 %

Using the binomial theorem of:

(p + q)^n = Σ nCr p^r q^(n-r) with r = 0, 1, ..., n

Each term gives the probability of r successes P(r) out of n trials: p is the probability of success and q = 1 - p is the probability of failure for each trial; for this question:

p = 1/5

q = 4/5

n = 10

and to get the probability of at least 6, P(≥ 6) is the sum P(6) + P(7) + P(8) + P(9) + P(10) which is:

10C6 p^6 q^4 + 10C7 p^7 q^3 + 10C8 p^8 q^2 + 10C9 p^9 q^1 + 10C10 p^10 q^0

= 210 x (1/5)^6 x (4/5)^4 + 120 x (1/5)^7 x (4/5)^3 + 45 x (1/5)^8 x (4/5)^2 + 10 x (1/5)^9 x (4/5)^1 + 1 x (1/5)^10 x (4/5)^0

= (210 x 1^5 x 4^4 + 120 x 1^7 x 4^3 + 45 x 1^8 x 4^2 + 10 x 1^9 x 4 + 1 x 1^10) x (1/5)^10

= 62201/9765625

≈ 0.0064 = 0.64 %