To answer this problem, we use the binomial distribution formula for probability:
P (x) = [n! / (n-x)! x!] p^x q^(n-x)
Where,
n = the total number of test questions = 10
x = the total number of test questions to pass = >6
p = probability of success = 0.5
q = probability of failure = 0.5
Given the formula, let us calculate for the probabilities that the student will get at least 6 correct questions by guessing.
P (6) = [10! / (4)! 6!] (0.5)^6 0.5^(4) = 0.205078
P (7) = [10! / (3)! 7!] (0.5)^7 0.5^(3) = 0.117188
P (8) = [10! / (2)! 8!] (0.5)^8 0.5^(2) = 0.043945
P (9) = [10! / (1)! 9!] (0.5)^9 0.5^(1) = 0.009766
P (10) = [10! / (0)! 10!] (0.5)^10 0.5^(0) = 0.000977
Total Probability = 0.376953 = 0.38 = 38%
There is a 38% chance the student will pass.