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ohn Jay is a major league pitcher. He pitches like a robot, that is he pitches the same way to each batter. There is a .56 probability that the first ball he pitches to a batter is a strike. (first-pitch strike). This is independent from batter to batter. Let X equal the number of first-pitch strikes John Jay throws in a game where he pitches to 15 batters. Let Y equal the number of batters he pitches to until he throws his 2nd first-pitch strike. (could be more than 15)

a. What is the probability X=13?
b. What is the standard deviation of X?
c. What is the probability that X > 10?
d. What is the probability that X ≥ 12?
e. What is the probability that Y = 3?
f. What is the probability that Y = 2?

Respuesta :

Answer:

Step-by-step explanation:

Given ;

  • Sample size, n = 15
  • probability of a first pitch strike, p = 0.56
  • Probability of not first pitch strike ; q = 1 - p = 1 - 0.56 = 0.44
  • Applying binomial distribution ; P(X = r) = nCr x P^r x (q)^n-r

a) P ( X = 13 ) = C(15,13) x 0.56^13 x (1-0.56)^2 = 0.0108

b) Mean = np = 15 x 0.56= 8.400

Standard deviation = √(np(1-p)) = √(15 x 0.56 x (1-0.56)) = 1.9225

c) P(X>10) = 1 - P(X≤10) = 1 - 0.8633 = 0.1367

d) P(X≥12) = P(X=12) + P(X=13) + P(X=14) + P(X=15) = 0.0498

e) P(Y=3) = C(3,1) x 0.56^1 x (1-0.56)^2 x 0.56 = 0.1821

f) P(Y=2) = C(2,1) 0.56^1 x (1-0.56)^1 x 0.56 = 0.2760