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# Probability of Detecting Defective and Non-Defective Items

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d) From the fitted line, it is clear that the next 3-point moving average will be 900

e) Inserting the predicted value in the formula for 3- point moving average we have;

(956+912+x)/3 =900

X = 832.

Thus, the value of sales for the first third of 2017 is 832

f) The answers in parts d and e are based on the estimated value from the line of best fit. They are not reliable since the line of best fit may have some errors in estimating the linear relationship between the two variables.

a) Taking the probability of passing the test to be 0.2, then the probability of passing at the 3rd attempt is given by 0.8* 0.8* 0.2 = 0.82 * 0.2

Generally, the prob. Of passing at the nth trial is given by (0.8n-1 * 0.2)

0.8n-1 * 0.2 < 1/500

0.8n-1

< 1/100

n-1 log 0.8 < log 0.01

n-1 > -2/-0.09691

ð n is approx. 22 attempts

b) The prob. of passing the test at nth trial is (0.9n-1 * 0.1)

ð (0.9n-1

* 0.1) < 1/50

0.9n-1

< 1/5

n-1 log 0.9 > log 0.2

n-1 > -0.69897/-0.04575749

n  >16.2757

ð n is approx. 17 attempts

c) In the formulation of the probability, it is assumed that every trial of passing the test is independent of one another. In real life, the assumption may not hold since previous interaction with the exam questions will influence the chance of an individual passing the test at nth trial.

a) Assuming a year with 365 days, I will need to meet (365/2) people sharing my birthday. That is, I will need to  meet about 183 people sharing my birthday after which it’s likely that the persons I meet don not share the same birthday as mine. The scenario is illustrated as follows; By meeting 2 people, the prob. of having different birthdays = 364/365 = 0. 99726.  If they are 3 people then the prob. of having different birthdays = (364/365) * (363/365) = 0.99726 * 0.99452 =0.9915.  The prob. that the four have different birthdays = (364/365) x (363/365) x (362/365) = 0.983327.

Since 23 people are needed in order to have two sharing their birthday with me, the prob. Is calculated by (364/365) x (363/365) x… x (343/365) = 0.493

Thus, the prob.that at least two people will be sharing their birthdays is given by 1- 0.493 = 0.507

b)

Let n= 8, 9, 10, and 11 respctively

Prob. Of defect not occurring is given by;

n=8, 1-p = 1-1/8 = 0.875

n=9, 1-p = 1-1/9 = 0.8889

n=10, 1-p = 1-1/10 = 0.9

n=11, 1-p = 1-1/11 = 0.90909

Prob. of not occurring in any of the wallpaper

Let n= 8, 9, 10, and 11 respctively

n=8, 1-p = (1-1/8)8

= 0.34361

n=9, 1-p = (1-1/9)9

= 0.346478

n=10, 1-p = (1-1/10)10

= 0. 348978

n=11, 1-p = (1-1/11)11 = 0.35049

Investigating further

Generally, the probability of get no defect in any of the wallpaper is represented as 1-p = (1-1/n)n

Summary and Conclusions

The line of best fit can be used in estimating the 3-point averages of the quarterly sales for the coming years without carrying out conclusions. However, since the line of best fit is an approximation, the forecasted values may not reflect the true amount of the quarterly sales. The assumption of independence occurrence of events can be used to formulate the probability of passing a given test after n number of trials. Besides, combination of normal and geometric population distributions helped in estimating the probability of detecting defective and non-defective items after a given number of trials. In conclusion, population distributions are essential for estimating the occurrence of independent events.

September 18, 2023
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526

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