Respuesta :
Answer:
He wont foul out with probability 0.9093
Step-by-step explanation:
The total number of fools he picked is a Binomial ditribution noted by X with parameters p = 0.05 and N = 48. The mean of this random variable is μ = np = 48*0.05 = 2.4 and the variance is σ² = np(1-p) = 2.4*0.95 = 2.28, hence its standard deviation is σ = √2.28 = 1.51.
Note that, if approximate probability is asked, we could just approximate X with a Normal random variable with mean 2.4 and standard deviation 1.51 (this can be done because of the central limit theorem). We will calculate the probability manually. He wont foul out if he picks 0,1,2,3 or 4 fouls, thus
[tex]P(X \leq 4) = P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4) \\= 0.95^{48}+48*0.95^{47}*0.05+{48 \choose 2}*0.95^{46}*0.05^2+\\{48 \choose 3}*0.95^{45}*0.05^3+{48 \choose 4}*0.95^{44}*0.05^4 \\= 0.085+0.215+0.266+0.215+0.127 = 0.9093[/tex]
As a consecuence, he wont foul out with probability 0.9093.
