Twelve trials are conducted in a Bernoulli process in which

Twelve trials are conducted in a Bernoulli process in which the probability of success in a given trial is 0.3. If x = the number of successes, determine the following:

Solution

P(X=0) = 12C0 * 0.3⁰ * 0.7¹² = 0.0138

P(X=1) = 12C1 * 0.3¹ * 0.7¹¹ = 0.0712

P(X=2) = 12C2 * 0.3² * 0.7¹⁰ = 0.1678

P(X=3) = 12C3 * 0.3³ * 0.7⁹ = 0.2397

P(X=4) = 12C4 * 0.3⁴ * 0.7⁸ = 0.2311

P(X=5) = 12C5 * 0.3⁵ * 0.7⁷ = 0.1585

P(X=6) = 12C6 * 0.3⁶ * 0.7⁶ = 0.0792

P(X=7) = 12C7 * 0.3⁷ * 0.7⁵ = 0.0291

P(X=8) = 12C8 * 0.3⁸ * 0.7⁴ = 7.7977 * 10^-3

P(X=9) = 12C9 * 0.3⁹ * 0.7³ = 1.4853 * 10^-3

P(X=10) = 12C10 * 0.3¹⁰ * 0.7² = 1.9096 * 10^-4

P(X=11) = 12C11 * 0.3¹¹ * 0.7¹ = 1.488 * 10^-5

P(X=12) = 12C12 * 0.3¹² * 0.7⁰ = 5.3144 * 10^-7

a) E(X) = 0 * 0.0138 + 1 * 0.0712 + 2 * 0.1678 + 3 * 0.2397 + 4 * 0.2311 + 5 * 0.1585 + 6 * 0.0792 + 7 * 0.0291 + 8 * 7.7977 * 10^-3 + 9 * 1.4853 * 10^-3 + 10 * 1.9096 * 10^-4 + 11 * 1.488 * 10^-5   + 12 * 5.3144 * 10^-7

= 3.6

b) E(X²) = 0² * 0.0138 + 1² * 0.0712 + 2² * 0.1678 + 3² * 0.2397 + 4² * 0.2311 + 5² * 0.1585 + 6² * 0.0792 + 7² * 0.0291 + 8² * 7.7977 * 10^-3 + 9² * 1.4853 * 10^-3 + 10² * 1.9096 * 10^-4 + 11² * 1.488 * 10^-5   + 12² * 5.3144 * 10^-7

= 15.48

var(X) = E(X²) - (E(X))²

           = 15.48 - 3.6²

          = 2.52

std dev = sqrt(2.52) = 1.587

c) P(X=3) = 12C3 * 0.3³ * 0.7⁹ = 0.2397

d) P(2 < X < 8 ) = P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8)

                         = 0.1678 + 0.2397 + 0.2311 + 0.1585 + 0.0792 + 0.0291 + 7.7977 * 10^-3

                         = 0.9132

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

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

                  = 1 - (0.0138 + 0.0712 + 0.1678 + 0.2397 )

                 = 1 - 0.4925 = 0.5075