A student makes random guesses for the the ten true-false question on a quiz.Find the probability that there i

is at least one corect answer.A student makes random guesses for the the ten true-false question on a quiz.Find the probability that there i
Another way of wording this is that they're not all wrong. So you're looking for 1 - P(all wrong), which is 1 - (1/2)^10.A student makes random guesses for the the ten true-false question on a quiz.Find the probability that there i
Let X be the number of correct answers. X has the binomial distribution with n = 10 trials and success probability p = 0.5





In general, if X has the binomial distribution with n trials and a success probability of p then


P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)


for values of x = 0, 1, 2, ..., n


P[X = x] = 0 for any other value of x.





The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.


Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.





X ~ Binomial( n = 10 , p = 0.5 )





the mean of the binomial distribution is n * p = 5


the variance of the binomial distribution is n * p * (1 - p) = 2.5


the standard deviation is the square root of the variance = 鈭?( n * p * (1 - p)) = 1.581139





The Probability Mass Function, PMF,


f(X) = P(X = x) is:





P( X = 0 ) = 0.0009765625


P( X = 1 ) = 0.009765625


P( X = 2 ) = 0.04394531


P( X = 3 ) = 0.1171875


P( X = 4 ) = 0.2050781


P( X = 5 ) = 0.2460938


P( X = 6 ) = 0.2050781


P( X = 7 ) = 0.1171875


P( X = 8 ) = 0.04394531


P( X = 9 ) = 0.009765625


P( X = 10 ) = 0.0009765625








The Cumulative Distribution Function, CDF,


F(X) = P(X 鈮?x) is:





x


鈭?P(X = t) =


t = 0





P( X 鈮?0 ) = 0.0009765625


P( X 鈮?1 ) = 0.01074219 = 1- P(X = 0) %26lt;%26lt;%26lt; ANSWER


P( X 鈮?2 ) = 0.0546875


P( X 鈮?3 ) = 0.171875


P( X 鈮?4 ) = 0.3769531


P( X 鈮?5 ) = 0.6230469


P( X 鈮?6 ) = 0.828125


P( X 鈮?7 ) = 0.9453125


P( X 鈮?8 ) = 0.9892578


P( X 鈮?9 ) = 0.9990234


P( X 鈮?10 ) = 1
P(at least 1 correct) = 1 - P(none are correct)





the probability that none of the 10 answers are correct is (1/2)^10





P = 1 - (1/2)^10


P 鈮?0.9990234375





edit:


1023/1024 鈮?0.9990234375


so the answer is A
so the only exception to at least one right is all wrong. The probability of this is (1/2)^10. You subtract this from 1 to since it is what we don't want, and the probability is 1023/1024
P = 1/2^10





= 0.0009765625