Question

You are an accountant who is performing the annual audit of the accounts at Hookem, Billem, and Soakem, a large cellular telephone company. From past experience it is known that 10% of the accounts at Hookem, Billem, and Soakem have errors.

Use this Excel file geometric probabilities 2 to assist you with the geometric probability calculations required to answer the questions below.

Question 1. What is the probability that the first account with an error is the 6th audited account?

(use 3 decimal places in your answer)

Question 2. What is the expected number of accounts that you will audit until you find the first error?

(use 2 decimal places in your answer)

Question 3. What is the probability that the first account with an error is among the first 6 accounts that you check?

(use 3 decimal places in your answer)

p

0.09

x

P(X <= x)

P(X = x)

1

0.0900

0.0900

2

0.1719

0.0819

3

0.2464

0.0745

4

0.3143

0.0678

5

0.3760

0.0617

6

0.4321

0.0562

7

0.4832

0.0511

8

0.5297

0.0465

9

0.5721

0.0423

10

0.6106

0.0385

11

0.6456

0.0350

12

0.6775

0.0319

13

0.7065

0.0290

14

0.7330

0.0264

15

0.7570

0.0240

16

0.7789

0.0219

17

0.7988

0.0199

18

0.8169

0.0181

19

0.8334

0.0165

20

0.8484

0.0150

21

0.8620

0.0136

22

0.8744

0.0124

23

0.8857

0.0113

24

0.8960

0.0103

25

0.9054

0.0094

26

0.9139

0.0085

27

0.9216

0.0078

28

0.9287

0.0071

29

0.9351

0.0064

30

0.9409

0.0058

31

0.9463

0.0053

32

0.9511

0.0048

33

0.9555

0.0044

34

0.9595

0.0040

35

0.9631

0.0036

36

0.9665

0.0033

37

0.9695

0.0030

38

0.9722

0.0027

39

0.9747

0.0025

40

0.9770

0.0023

41

0.9791

0.0021

42

0.9810

0.0019

43

0.9827

0.0017

44

0.9842

0.0016

45

0.9856

0.0014

46

0.9869

0.0013

47

0.9881

0.0012

48

0.9892

0.0011

49

0.9902

0.0010

50

0.9910

0.0009

Answer 1

Given  

p = 10% = 0.10

q = 1 - 0.10 = 0.90

a)

The probability that the first account with an error is the 6th audited account

= p( first 5 are correct and 6^th has error )

= (0.10) (0.90)^6-1

= (0.10) (0.90)⁵

= 0.05905  0.059

b)

The expected number of accounts that you will audit until you find the first error

= 1 / 0.10 = 10.00

c)

the probability that the first account with an error is among the first 6 accounts that you check

= 1 - p (till first 6 no error account is found)

= 1 - (0.90)⁶

= 0.468

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