Question

At your college 30 % of all students are from out of state. If you randomly select 15 of these students, what is the probability that: A. All are from within the state? B. All from out of state? C. Exactly two are from within the state? D. At least 5 are from within the state?

Please show how to work out the problem, it helps a lot. Thank you!

Answer 1

X ~ Bin(n,p)

Where n = 15

P(Out of state) = 0.30 , P(Within state) = 1 - 0.30 = 0.70

P(X) = ⁿC_x * p^x ( 1 - p)^n-x

a)

P(Within state) = 1 - 0.30 = 0.70

P(X = 15) = ¹⁵C₁₅ * 0.70¹⁵ * 0.30⁰

= 0.0047

b)

P(Out of state) = 0.30

P(X = 15) = ¹⁵C₁₅ * 0.30¹⁵ * 0.70⁰

= 0.0000

c)

P(Within state) = 1 - 0.30 = 0.70

P(X = 2) = ¹⁵C₂ * 0.70² * 0.30¹³

= 0.00001

d)

P(Within state) = 1 - 0.30 = 0.70

P(X >= 5) = 1 - P(X <= 4)

= 1 - 0.00067 [Probability calculated from Cumulative binomial probability table with p = 0.70 , n = 15 , x = 4 )

= 0.9993