Question

A survey shows that 30% of adults own stocks and 40% own mutual funds. A sample of 10 adults is chosen. Use binomial distribution to answer the following questions.

(a) (7 points) What is the probability that there will be less than 2 adults owning mutual funds?

            Answer: ___________________________

(b) (6 points) Determine the probability that there will be at least 2 adults owning stocks.

            Answer: __________________________________

(c) (6 points) Determine the variance of the number of adults not owning stocks.

            Answer: ________________________________

(d) (6 points) What is the probability of getting 5 or 6 adults owning mutual funds?

           

            Answer: ___________________________

Answer 1

a)

Here, n = 10, p = 0.4, (1 - p) = 0.6 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X < 2).
P(X <2) = (10C0 * 0.4^0 * 0.6^10) + (10C1 * 0.4^1 * 0.6^9)
P(X < 2) = 0.006 + 0.0403
P(X <2) = 0.0463

b)

Here, n = 10, p = 0.3, (1 - p) = 0.7 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X >= 2).
P(X >= 2) = (10C2 * 0.3^2 * 0.7^8) + (10C3 * 0.3^3 * 0.7^7) + (10C4 * 0.3^4 * 0.7^6) + (10C5 * 0.3^5 * 0.7^5) + (10C6 * 0.3^6 * 0.7^4) + (10C7 * 0.3^7 * 0.7^3) + (10C8 * 0.3^8 * 0.7^2) + (10C9 * 0.3^9 * 0.7^1) + (10C10 * 0.3^10 * 0.7^0)
P(X >= 2) = 0.2335 + 0.2668 + 0.2001 + 0.1029 + 0.0368 + 0.009 + 0.0014 + 0.0001 + 0
P(X >= 2) = 0.8506

c)

varaince = npq

= 10 * 0.40 * 0.60
= 2.40

d)

Here, n = 10, p = 0.4, (1 - p) = 0.6 and x = 5
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 5)
P(X = 5) = 10C5 * 0.4^5 * 0.6^5
P(X = 5) = 0.2007

Here, n = 10, p = 0.4, (1 - p) = 0.6 and x = 6
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 6)
P(X = 6) = 10C6 * 0.4^6 * 0.6^4
P(X = 6) = 0.1115

P( 5 or 6 ) = 0.2007 + 0.1115
= 0.3123