What's the probability of randomly guessing 8 correct answers out of 10 on a True and False test?

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 , p )





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


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








1 - F(X) is:





n


鈭?P(X = t) =


t = x





P( X 鈮?0 ) = 1


P( X 鈮?1 ) = 0.9990234


P( X 鈮?2 ) = 0.9892578


P( X 鈮?3 ) = 0.9453125


P( X 鈮?4 ) = 0.828125


P( X 鈮?5 ) = 0.6230469


P( X 鈮?6 ) = 0.3769531


P( X 鈮?7 ) = 0.171875


P( X 鈮?8 ) = 0.0546875


P( X 鈮?9 ) = 0.01074219


P( X 鈮?10 ) = 0.0009765625What's the probability of randomly guessing 8 correct answers out of 10 on a True and False test?
2/5What's the probability of randomly guessing 8 correct answers out of 10 on a True and False test?
I don't know if this helps, but I got something close to the answer.





(1/2) / 10 = .05 [one-half divided by ten equals five-hundredths]





Since there are only two choices on the test (True or False), you automatically have a 50% (1/2) chance of getting a question right.





There are 10 questions on the test.
We have to use the binomial distribution:





[(Probability Correct ^ # Correct) x (Probability Wrong ^ # Wrong)} * [# Questions! / # Correct!# Wrong!]





That last part with the ! is finding the total number of combinations of a given number of right and wrong you can have, the ! is a factoral, should be on your calculator.





So for 8 correct, in any order:





[(0.50 ^ 8) x (0.50 ^ 2)] * [10! / 2!8!] = 0.000977 * 45 = 4.40%





Now if it can be 8, 9, or 10 correct, then you are looking at adding in the probability of 9 or 10 correct:





[(0.50 ^ 9) x (0.50 ^ 1)] * [10! / 1!9!] = 0.000977 * 10 = 0.98%


(0.50 ^10) = 0.098%





So: 4.40% + 0.98% + 0.098% = 5.5%