(2003) studied the number of goals scored during 232 World Cup soccer games play

Question

(2003) studied the number of goals scored during 232 World Cup soccer games played from 1990 to 2002 and found them to be well modeled by a Poisson random variable (x). Only goals during the 90 minutes of regulation play were discovered. The average number of goals scored each game (i.e lambda) was 2.5. Assuming that value continues to hold for other World Cup games, find the probability of the following events: a) at least 4 goals are scored in negative play in a randomly selected game in the next World Cup b) no goals are scored in regulated play in a upcoming World Cup game. c) at most 2 goals as scored in play. A manufacturer has three machines producing light Machine A produces 40% of the light bulbs with 1% of them defective. Machine B produces 35% of them with 2% defective. Machine C produces 25% of the light bulbs with 4% being defective. (i) What is the probability of selecting (randomly) a defective bulb. (ii) If a lightbulb is tested and found to be defective. What is the probability that it was produced by machine C? (iii) If a light bulb is tested to be defective What is the probability that it was produced by machine A?

Explanation / Answer

Given,

P(A)= 0.4

P(B)=0.35

P(C)=0.25

P(D/A)=0.01

P(D/B)=0.02

P(D/C)=0.04

A.a) probability of selectig a defective bulb is

P(D)= 0.4*0.01+0.35*0.02+0.25*0.04

P(D)=0.021

A.b) to find P(C/D)=0.25*0.04/0.4*0.01+0.35*0.02+0.25*0.04

solving we get

Prob of defective bulb produced by C is 0.476

A.c) To find P(A/D)=0.4*0.01/0.4*0.01+0.35*0.02+0.25*0.04

solving we get

Prob of defective bulb produced by A is 0.19

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