13. Suppose you had to guess on a four - choice multiple - choice test and were given four questions . Find the binomial probability distribution . n = (p + q) ^ n when 4 and p = 0.25
p^4 + 4 p^3 q + 6 p^2 q^2 + 4 p q^3 + q^4
To find the binomial probability distribution for a four-choice multiple-choice test, where n = 4 and p = 0.25, we need to calculate the probability of getting each possible number of correct answers.
The formula for the binomial probability distribution is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where P(X=k) is the probability of getting k correct answers out of n questions, C(n,k) is the number of ways to choose k questions out of n, p is the probability of getting a question correct, and (1-p) is the probability of getting a question incorrect.
Let's calculate the probability for each possible number of correct answers (k=0, 1, 2, 3, 4) using the given values:
For k = 0:
P(X=0) = C(4,0) * (0.25)^0 * (1-0.25)^(4-0) = 1 * 1 * 0.75^4 = 0.3164
For k = 1:
P(X=1) = C(4,1) * (0.25)^1 * (1-0.25)^(4-1) = 4 * 0.25 * 0.75^3 = 0.4219
For k = 2:
P(X=2) = C(4,2) * (0.25)^2 * (1-0.25)^(4-2) = 6 * 0.25^2 * 0.75^2 = 0.2813
For k = 3:
P(X=3) = C(4,3) * (0.25)^3 * (1-0.25)^(4-3) = 4 * 0.25^3 * 0.75^1 = 0.0938
For k = 4:
P(X=4) = C(4,4) * (0.25)^4 * (1-0.25)^(4-4) = 1 * 0.25^4 * 0.75^0 = 0.0039
Therefore, the binomial probability distribution for this four-choice multiple-choice test with p = 0.25 is as follows:
P(X=0) = 0.3164
P(X=1) = 0.4219
P(X=2) = 0.2813
P(X=3) = 0.0938
P(X=4) = 0.0039