Probability question.. stat expert help !!?

Question

Each online assignment in our class has 20 multiple choice questions, each has five possible answers and only one answer is correct.

a) If you randomly guess all the answers, what is the probability that none of the answers is correct?

Answer 1

Let X be the number of correct answers. X has the binomial distribution with n = 20 trials and success probability p = 0.2



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 = 20 , p = 0.2 )



the mean of the binomial distribution is n * p = 4

the variance of the binomial distribution is n * p * (1 - p) = 3.2

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 1.788854



The Probability Mass Function, PMF,

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



P( X = 0 ) = 0.01152922 ← answer

P( X = 1 ) = 0.05764608

P( X = 2 ) = 0.1369094

P( X = 3 ) = 0.2053641

P( X = 4 ) = 0.2181994

P( X = 5 ) = 0.1745595

P( X = 6 ) = 0.1090997

P( X = 7 ) = 0.05454985

P( X = 8 ) = 0.02216088

P( X = 9 ) = 0.007386959

P( X = 10 ) = 0.002031414

P( X = 11 ) = 0.0004616849

P( X = 12 ) = 8.656592e-05

P( X = 13 ) = 1.331783e-05

P( X = 14 ) = 1.664729e-06

P( X = 15 ) = 1.664729e-07

P( X = 16 ) = 1.300570e-08

P( X = 17 ) = 7.65041e-10

P( X = 18 ) = 3.187671e-11

P( X = 19 ) = 8.388608e-13

P( X = 20 ) = 1.048576e-14

Answer 2

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Answer 3

The simple answer to this: since getting one question wrong is independent of any other, the probability of getting all of them wrong is simply:



(4/5)^20 ~= 0.0115

Answer 4

If you guess at one problem, you will have a 80%, or 0.80 chance of getting it wrong. Since each guess is independent, we multiply the probabilities together to get the probability that all will be wrong. 0.80 multiplied by itself 20 times is 0.8^20. Calculator time.

Answer 5

4/5 ^ 20 = 0.0115 (1.15%)

Answer 6

.8^20

=1.15%

Answer 7

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