Answer:
a) f(12) = P(X=12) = 0.1144
b) f(16) = P(X=16) = 0.1304
c) P(x≥16) = 0.2375
d) P(x≤15) = 0.7625
e) E(X) = 14
f) Var(X) = 4.2
Standard deviation = 2.05
Step-by-step explanation:
Binomial distribution formula is given as
Binomial distribution function is represented by
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
n = total number of sample spaces = 20
x = Number of successes required = variable
p = probability of success = 0.7
q = probability of failure = 1 - 0.7 = 0.3
a) f(12) = P(X=12)
x = 12
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
P(X = 12) = ²⁰C₁₂ (0.7)¹² (0.3)²⁰⁻¹²
P(X=12) = 0.1144
b) f(16) = P(X=16)
x = 16
P(X = 16) = ²⁰C₁₆ (0.7)¹⁶ (0.3)²⁰⁻¹⁶
P(X=16) = 0.1304
c) P(x≥16)
This is a sum of probabilities from x = 16 to x = 20. (Total number of sample space = 20)
P(x≥16) = P(X=16) + P(X=17) + P(X=18) + P(X=19) + P(X=20) = 0.2375 (computed using calculator)
d) P(x≤15)
This is a sum of probabilities from 0 to 15.
But it can be rewritten as
P(X≤15) = 1 - P(X > 15) = 1 - [P(X=16) + P(X=17) + P(X=18) + P(X=19) + P(X=20)]
= 1 - 0.2375 = 0.7625
e) Expected value = E(X) = Mean = np = (20)(0.7) = 14
f) Variance = np(1-p) = (20×0.7×0.3) = 4.2
Standard deviation = σ = √(variance) = √(4.2) = 2.05
Hope this Helps!!!
