If 𝑥 is a binomial random variable, compute 𝑝(𝑥) for each of the following cases:?
a) 𝑛=7,𝑥=5,𝑝=0.6
b) 𝑛=10,𝑥=7,𝑝=0.3
(c) 𝑛=8,𝑥=3,𝑝=0.5
(d) 𝑛=3,𝑥=0,𝑝=0.7
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a) 𝑛=7,𝑥=5,𝑝=0.6
b) 𝑛=10,𝑥=7,𝑝=0.3
(c) 𝑛=8,𝑥=3,𝑝=0.5
(d) 𝑛=3,𝑥=0,𝑝=0.7
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Verified answer
Use the formula
P( of exactly x successes in n trials = (n x ) p^x (1-p)^(x-k)
where
(n x ) = n! / (x! (n-x)! )
a) 𝑛=7,𝑥=5,𝑝=0.6
so first
(n x) = 7! / (5! 2! ) = 7*6/2 = 21
P(exactly 5 successes) = 21*(0.6)^5 (0.4)^2 = 0.2612736
(checks against spreadsheet function)
b) 𝑛=10,𝑥=7,𝑝=0.3
so first
(10 7) = 10! / ( 7! *3!) = 10*9*8/ 3! = 10*9*8/6 = 10*3*8/2 = 10*3*4 = 120
p(exactly 5 successes) = 120*( 0.3)^7*(0.7)^3 = 0.009001692
(checks against spreadsheet function)
(c) 𝑛=8,𝑥=3,𝑝=0.5
first
(8 3) = 8! / ( 5! 3!) = 8*7*6/3*2*1 = 8*7 = 56
p(exactly 3 successes) = 56*(0.5)^3*(0.5)^5 = 56*(0.5)^8
P(exactly 3 successes ) = 56*(0.5)^8 = 0.21875
(checks against spreadsheet function)
(d) 𝑛=3,𝑥=0,𝑝=0.7
(3 0 ) = 3! / (0! 3!) = 1
P(exactly 0) = 1*(0.7)^(0) *(0.3)^3 = 0.3^3 = 0.027
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