Question

38​% of a certain​ country's voters think that it is too easy to vote in their country. You randomly select 12 likely voters. Find the probability that the number of likely voters who think that it is too easy to vote is​

(a) exactly​ three, (b) at least​ four, (c) less than eight.

​(a) P(3)=   ​(Round to three decimal places as​ needed.)

​(b) Upper P left parenthesis x greater than or equals 4 right parenthesis equals nothing ​(Round to three decimal places as​ needed.)

​(c) Upper P left parenthesis x less than 8 right parenthesis equals nothing ​(Round to three decimal places as​ needed.)

Answer 1

X ~ Binomial (n,p)

Where n = 12 , p = 0.38

P(X) = ⁿC_x p^x ( 1 - p)^n-x

a)

P( X = 3) = ¹²C₃0.38³ 0.62⁹

= 0.163

b)

P( X >= 4 ) = 1 - P( X <= 3)

= 1 - [ P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) ]

= 1 - [ ¹²C₀0.38⁰ 0.62¹² +¹²C₁0.38¹ 0.62¹¹ +¹²C₂0.38² 0.62¹⁰ +¹²C₃0.38³ 0.62⁹ ]

= 1 - 0.2704

= 0.730

P( X < 8) = 1 - P( X >= 8)

= 1 - [ P( X = 8) + P( X = 9) + P( X + 10) + P( X = 11) + P( X = 12) ]

= 1 - [ ¹²C₈0.38⁸ 0.62⁴ +¹²C₉0.38⁹ 0.62³ +¹²C₁₀0.38¹⁰ 0.62² +¹²C₁₁0.38¹¹ 0.62¹ +¹²C₁₂0.38¹² 0.62⁰ ]

= 1 - 0.0422

= 0.958