A survey indicates that 41 of women in the United States con

A survey indicates that 41% of women in the United States consider reading their favorite leisure-time activity. You randomly select four U.S. women and ask them if reading is their favorite leisure-time activity. Find the probability that: a. exactly two of them respond yes b. at least two of them respond yes c. fewer than two of them respond yes

Solution

Probability that women in United states consider reading as their favourite leisure-time activity i.e p = 0.41

q = Probability that women in United states does not consider reading as their favourite leisure-time activity = 1 - 0.41 = 0.59

No. of Womens in United states consider reading as their favourite leisure-time activity ~ B(5, 0.41)

a) P(X = 2) = ⁵C₂ * (0.41)² * (0.59)³ = 10 * 0.1681* 0.2054 = 0.3452

Probability that exactly two women respond yes = 0.3452

b) P(X>=2) = 1- P(X<2) = 1 - {⁵C₀ * (0.59)⁵ + ⁵C₁ * (0.41)¹ * (0.59)⁴}

= 1 - {0.0715 + (5 * 0.41 * 0.1212)} = 1 - {0.0715 + 0.24846} = 1- 0.31996 = 0.68004

Probability that atleast two women respond yes = 0.68004

c) P(X<2) is already computed in b = {⁵C₀ * (0.59)⁵ + ⁵C₁ * (0.41)¹ * (0.59)⁴ } = 0.31996

Probability that less than two women respond yes = 0.31996