6.16 Calculate each binomial probability:
a. X = 2, n = 8, ð = .10
b. X = 1, n = 10, ð = .40
c. X = 3, n = 12, ð = .70
d. X = 5, n = 9, ð = .90
To calculate the binomial probability, you can use the formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
where:
- n is the number of trials
- k is the number of successful outcomes
- p is the probability of success
Now let's calculate each binomial probability:
a. X = 2, n = 8, ð = .10
Using the formula, substitute the values:
P(X = 2) = 8C2 * (0.10)^2 * (1-0.10)^(8-2)
Calculate 8C2 = 28:
P(X = 2) = 28 * (0.10)^2 * (1-0.10)^6
Calculate (0.10)^2 = 0.01 and (1-0.10)^6 = 0.531441:
P(X = 2) = 28 * 0.01 * 0.531441
Multiply the values together:
P(X = 2) = 0.14925344
b. X = 1, n = 10, ð = .40
Using the formula, substitute the values:
P(X = 1) = 10C1 * (0.40)^1 * (1-0.40)^(10-1)
Calculate 10C1 = 10:
P(X = 1) = 10 * (0.40)^1 * (1-0.40)^9
Calculate (0.40)^1 = 0.40 and (1-0.40)^9 = 0.262144:
P(X = 1) = 10 * 0.40 * 0.262144
Multiply the values together:
P(X = 1) = 1.0485760
c. X = 3, n = 12, ð = .70
Using the formula, substitute the values:
P(X = 3) = 12C3 * (0.70)^3 * (1-0.70)^(12-3)
Calculate 12C3 = 220:
P(X = 3) = 220 * (0.70)^3 * (1-0.70)^9
Calculate (0.70)^3 = 0.343 and (1-0.70)^9 = 0.001572864:
P(X = 3) = 220 * 0.343 * 0.001572864
Multiply the values together:
P(X = 3) = 0.12089513
d. X = 5, n = 9, ð = .90
Using the formula, substitute the values:
P(X = 5) = 9C5 * (0.90)^5 * (1-0.90)^(9-5)
Calculate 9C5 = 126:
P(X = 5) = 126 * (0.90)^5 * (1-0.90)^4
Calculate (0.90)^5 = 0.59049 and (1-0.90)^4 = 0.0001:
P(X = 5) = 126 * 0.59049 * 0.0001
Multiply the values together:
P(X = 5) = 0.007075176
Therefore, the calculated probabilities are:
a. P(X = 2) = 0.14925344
b. P(X = 1) = 1.0485760
c. P(X = 3) = 0.12089513
d. P(X = 5) = 0.007075176