Question

Find the indicated probabilities using the geometric​ distribution, the Poisson​ distribution, or the binomial distribution. Then determine if the events are unusual. If​ convenient, use the appropriate probability table or technology to find the probabilities. Sixty dash two percent of U.S. adults oppose hydraulic fracturing​ (fracking) as a means of increasing the production of natural gas and oil in the United States. You randomly select six U.S. adults. Find the probability that the number of U.S. adults who oppose fracking as a means of increasing the production of natural gas and oil in the United States is​ (a) exactly three​, ​(b) less than four​, and​ (c) at least three.

Answer 1

X ~ Binomial (n,p)

Where n = 6 , p = 0.62

P(X) = ⁿC_x p^x (1 - p)^n-x

a)

P( X = 3) = ⁶C₃ 0.62³ 0.38³

= 0.2616

b)

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

= 1 - [ P (X = 4) + P( X = 5) + P( X = 6) ]

= 1 - [ ⁶C₄ 0.62⁴ 0.38² + ⁶C₅ 0.62⁵ 0.38 + ⁶C₆ 0.62⁶ 0.38⁰ ]

= 1 - 0.5857

= 0.4143

c)

P( X >= 3) = 1 - P( X <= 2)

= 1 - [ P (x = 0) + P( X = 1 ) + P( X = 2) ]

= 1 - [ ⁶C₀ 0.62⁰ 0.38⁶ + ⁶C₁ 0.62¹ 0.38⁵ + ⁶C₂ 0.62² 0.38⁴ ]

= 0.8473

The event is unusual if probability of occurring event is less than 0.05

Since all probabilities are grater than 0.05, the events are not unusual.