Suppose that X is binomial (4, 0.5), find P[X>2]
Would the answer be 0.5?
To find P[X>2], we need to calculate the probability of observing more than 2 successes in a binomial distribution with parameters n = 4 and p = 0.5.
The probability mass function (PMF) of a binomial distribution is given by P(X = k) = (n C k)(p^k)((1-p)^(n-k)), where (n C k) is the binomial coefficient: n! / (k!(n-k)!)
In this case, we want to find P[X > 2], which includes the probabilities for X = 3 and X = 4. Therefore, we need to sum the probabilities for these two values:
P[X > 2] = P[X = 3] + P[X = 4]
Using the PMF formula and substituting the values, we get:
P[X > 2] = (4 C 3)(0.5^3)(0.5^(4-3)) + (4 C 4)(0.5^4)(0.5^(4-4))
Calculating each term:
P[X > 2] = (4 C 3)(0.5^3)(0.5^1) + (4 C 4)(0.5^4)(0.5^0)
= 4(0.5^3)(0.5) + 1(0.5^4)(1)
= 4(0.125)(0.5) + 1(0.0625)(1)
= 0.25 + 0.0625
= 0.3125
Therefore, P[X > 2] is 0.3125, not 0.5.