Suppose that a table of part counts is generalized as

Chapter 2, Problem 235MEE

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QUESTION:

Suppose that a table of part counts is generalized as follows:

where 𝑎, 𝑏, and 𝑘 are positive integers. Let 𝐴 denote the event that a part is from supplier 1, and let 𝐵 denote the event that a part conforms to specifications. Show that 𝐴 and 𝐵 are independent events. This exercise illustrates the result that whenever the rows of a table (with 𝑟 rows and 𝑐 columns) are proportional, an event defined by a row category and an event defined by a column category are independent.

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QUESTION:

Suppose that a table of part counts is generalized as follows:

where 𝑎, 𝑏, and 𝑘 are positive integers. Let 𝐴 denote the event that a part is from supplier 1, and let 𝐵 denote the event that a part conforms to specifications. Show that 𝐴 and 𝐵 are independent events. This exercise illustrates the result that whenever the rows of a table (with 𝑟 rows and 𝑐 columns) are proportional, an event defined by a row category and an event defined by a column category are independent.

ANSWER:

Solution:

Step 1 of 2:

We have,

                                      Conforms

Suppliers

Yes

No

Total

1

ka

kb

K(a+b)

2

a

b

a+b

Total

Ka + a

Kb + b

Where, a, b, and k are positive integers.

Let, A = part from supplier 1.

       B = part conforms to specifications.

AB = events of part supplied by supplier 1 are confirms.

The claim is to show that the event defined by a column category are independent.


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