Math statistic work

Question

A survey indicates that 40% of women in U.S. consider reading their favorite leisure - time activity. You randomly select 10 U.S. women And ask them is reading their favorite leisure- time activity.

1. Find the probability exactly two of them respond yes?

2. Find the probability at least two of them respond yes?

Answer 1

Hello Jason,

Let X = (Number of women in a random sample of 10 US women who say reading is their favorite leisure activity.) With the information you provide, we may reasonably assume that X follows a binomial distribution. The important parameters for a binomial distribution are n (the number of "trials") and p (the probability of "success" on any one trial.) For this situation, we have p = .4 and n = 10. The formula for the binomial distribution is

P(x) = _nC_x·p^x·(1 - p)^n-x

Here, x is the number of "successes" in n "trials", or, in this problem, the number of women in a random sample of 10 who say reading is their favorite leisure activity. _nC_x is the "number of combinations of n things taken x at a time". This is the number ways of choosing x distinct objects from a set of n objects, without regard to order. It is given by

_nC_x = n!/[x!·(n-x)!]

Here, ! is the factorial symbol.

Part 1. Find the probability that exactly 2 of them respond yes.

Use the above formula with x = 2.

P(2) = ₁₀C₂·(.4)²(.6)^(10-2)

Calculating ₁₀C₂, we have ₁₀C₂ = 10!/(2!·8!) = 45.

Thus,

P(2) = 45(.4)²(.6)⁸ = .1209 (to four decimal places.)

Part 2. Find the probability that at least two of them respond yes.

In the problem at hand, "at least 2" means X≥2. This could be calculated by adding the probabilities for X=2, X=3, ..., X=10. That is P(X≥2) = P(2) + P(3) + P(4) + ... + P(10). This is rather tedious if you have to do the calculation by hand. It is easier to calculate the probability of the complement of the event and subtract from one. The complement of the event "At least two (X≥2)" is "Less than 2 (X<2)". Since X is discrete, "X<2" is equivalent to "X=0 or X=1".

P(At least 2) = P(X≥2) = 1 - P(X<2)

= 1 - (P(0) + P(1)).

Use the binomial distribution formula from above to find P(0) and P(1). We obtain P(0) = .00605 and P(1) = .04031 (details omitted.) Therefore,

P(At Least 2) = 1 - (.00605 + .04031)

= 1 - .04636

= .95364

Hope that helps! If you need any additional explanation, please let me know.

William

(Available online for Statistics tutoring.)