Question

Write by hand Q1&2，need to show step by step

For each of the following questions: clearly indicate the probability distribution being used to solve the problem solve by hand, and verify your answer using MATLAB. 1. Two teams, A and B, play a series of games. If team B has a probability 0.4 of winning each game, is it to their advantage to play the best three out of five games or the best four out of seven, and why? Assume the outcomes of successive games are independent. 2. A quality control engineer works at a factory that bottles engine oil. The amount of oil is normally distributed with mean 40 litres and standard deviation of 0.05 litres. If the amount is less than 39.9 litres or more than 40.1 litres it is deemed to fail the quality control process. (a) Find the probability that a bottle will fail quality control (b) If the factory want to improve so that a bottle will fail 2% of the time, what must the quality control amounts be extended to?

Answer 1

1)

Here the probability of winning for B is 0.4.

The probabiility fo team B winning the best of three out of five games is

$P(X>=3) = P(X=3) + P(X=4)+P(X=5) = \binom{5}{3}0.4^{3} 0.6^{2}+ \binom{5}{4}0.4^{4} 0.6^{1} + \binom{5}{5}0.4^{5} 0.6^{0} = 0.2304+0.0768 +0.0102 = 0.3174$

Out of 7 games,

$P(X>=4) = P(X=4) + P(X=5)+P(X=6) +P(X=7) = \binom{7}{4}0.4^{4} 0.6^{3}+ \binom{7}{5}0.4^{5} 0.6^{3} + \binom{7}{6}0.4^{6} 0.6^{1} + \binom{7}{7}0.4^{7} 0.6^{0} = 0.1935 +0.0774+ 0.0172+ 0.0016 = 0.2897$

Since, the probability of winning is higher in the first case, team B should go with that

2)

a)

P(X<39.9 or X>40.1) = 1- P(39.9<X<40.1)

µ =    40                              
σ =    0.05
we need to calculate probability for ,                                  
39.9   ≤ X ≤    40.1                          
X1 =    39.9   ,   X2 =   40.1                  
                                  
Z1 =   (X1 - µ ) / σ =   -2.000                          
Z2 =   (X2 - µ ) / σ =   2.000                          
                                  
P (   39.9   < X <    40.1   ) =    P (    -2   < Z <    2.000   )
                                  
= P ( Z <    2.000   ) - P ( Z <   -2.000   ) =    0.9772   -    0.0228   =    0.9545
so, Probability that a bottle will fail quality control = 1 - P (   39.9   < X <    40.1   ) = 1-0.9545 = 0.0455

b)

Given that

P(X<a or X >b)=0.02

P(a<x<b)=1-0.01=0.98

P(-2.33<Z<2.33)=0.98 [ excel formula =NORMSINV(0.98)

NOW,

(x-µ)/σ=±2.33

(X-40)/0.05=±2.33

X=39.88, 40.12

so, a=39.88

b=40.12

so, the amount should be either less than 39.88 or more than 40.12(answer)