STATS HELP?!???? ----Randomly guessing in a true false quiz and there are four questions?

wat is the probabily that you get


none right?


exactly one right?


exactly two right?


exactly three right?


all four right?








can somebody show me how to do this...and how i would draw a tree diagram for this? step by stepp. thanks!STATS HELP?!???? ----Randomly guessing in a true false quiz and there are four questions?
Since the answers are true/false, when you guess, you have a 50% chance of getting each question right, and the same chance of getting it wrong.





There are 2*2*2*2 = 16 possible ways to answer the 4 questions. Here they are:





F F F F


F F F T


F F T F


F F T T


F T F F


F T F T


F T T F


F T T T


T F F F


T F F T


T F T F


T F T T


T T F F


T T F T


T T T F


T T T T





For the sake of discussion, assume the correct answer to each of the questions is true.





There's 1 way to get 0 right, so P(0) = 1/16





There are 4 ways to get 1 right, so P(1) = 4/16 = 1/4





There are 6 ways to get 2 right, so P(2) = 6/16 = 3/8





There are 4 ways to get 3 right, so P(3) = 4/16 = 1/4





There's 1 way to get 4 right, so P(4) = 1/16STATS HELP?!???? ----Randomly guessing in a true false quiz and there are four questions?
Let X be the number of correct answers. X has the binomial distribution with n = 4 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 = 4 , p = 0.5 )





the mean of the binomial distribution is n * p = 2


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


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





The Probability Mass Function, PMF,


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





P( X = 0 ) = 0.0625 %26lt;%26lt;%26lt; none right


P( X = 1 ) = 0.25 %26lt;%26lt;%26lt; exactly one right


P( X = 2 ) = 0.375 %26lt;%26lt;%26lt; exactly two right


P( X = 3 ) = 0.25 %26lt;%26lt;%26lt;%26lt; exactly three right


P( X = 4 ) = 0.0625 %26lt;%26lt;%26lt;%26lt; all four right
I hope u know what the Binomial probability distribution formula is... nCk(p)^k(1-p)^n-k








n= number of trials


k= number of success


p= probability





None correct: 4C0(1/2)^0(1/2)^4 = 1/16





1 correct : 4C1(1/2)^1(1/2)^3 = 1/4





2 correct: 4C2(1/2)^2(1/2)^2= 3/8





3 correct: 4C3(1/2)^3(1/2)^1= 1/4





4 correct: 4C4(1/2)^4(1/2)^0 = 1/16
P(correct) = 1/2


P(incorrect) = 1/2





P(none right) = P(four incorrect) = (1/2)^4 = 1/16





P(exactly 1 correct) = P(1 correct AND 3 incorrect) = 1/2 * (1/2)^3 = 1/16 * 4 ways = 4/16 = 1/4





P(exactly 2 correct) = P(2 correct AND 2 incorrect) = (1/2)^2 *(1/2)^2 = 1/16 * 6 ways = 6/16 = 3/8





P(exactly 3) = 1/16 * 4 ways = 4/16 = 1/4





P(all 4 correct) = 1/16