Can you please help me with this

Question

A quiz consists of 10 multiple-choice questions, each with 6 possible answers. For someone who makes random guesses for all of the answers, find the probability of passing if the minimum passing grade is 60 %.

P(pass)=�(pass)= 

(Can you please include calculations in each step)

Answer 1

The number of correct answers out of (n = 10) questions is a binomial random variable (denoted x) with probability of “success” on each trial (question) equal to (1÷6) and probability of “failure” equal to (5÷6). The binomial formula for calculating the probability of k successes P(x = k) in n identical trials with probability of success of each trial p and probability of failure on each trial q is P(x = k) = (n!)÷[k!(n-k)!]×p^k×q^n-k

Since the passing grade is 60% and there are 10 questions, the probability of passing the exam is the sum of the binomial probabilities of 6 through 10 successes. That is

P(pass) = P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + P(x = 10)

The individual binomial probabilities are calculated as follows:

P(x = 6) = (10!)÷(6!4!)×(1÷6)⁶×(5÷6)⁴≈ 0.00217

P(x = 7) = (10!)÷(7!3!)×(1÷6)⁷×(5÷6)³ ≈ 0.00248

P(x = 8) = (10!)÷(8!2!)×(1÷6)⁸×(5÷6)² ≈ 0.0000186

P(x = 9) = (10!)÷(9!1!)×(1÷6)⁹×(5÷6)¹ ≈ 8.27 × 10^-7

P(x = 10) = (10!)÷(10!0!)×(1÷6)¹⁰(5÷6)⁰ ≈ 1.65 × 10^-8

The sum of these probabilities should be about 0.00467