Question

Samples of emissions from three suppliers are classified for conformance to air-quality specifications. The results from 100 samples are summarized as follows:

conforms
yes no
1 22 8
supplier 2 25 5
3 30 10

Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample from any supplier conforms to specifications. If a disk is selected at random, determine the following probabilities.

a. P(A)=0.3
b. P(B)=0.77
c. P(A ∩ B) =0.22
d. P(A U B)=0.85

Answer 1

`P(A) = 0.30`

`P(B) = 0.77`

`P(A\ n\ B) = 0.22`

`P(A\ u\ B) = 0.85`

Step-by-step explanation:

Given

See attachment for proper data presentation

`n = 100` --- Sample

A = Supplier 1

B = Conforms to specification

Solving (a): P(A)

Here, we only consider data in sample 1 row.

Here:

`Yes = 22` and
`No = 8`

`n(A) = Yes + No`

`n(A) = 22 + 8`

`n(A) = 30`

P(A) is then calculated as:

`P(A) = (n(A))/(Sample)`

`P(A) = (30)/(100)`

`P(A) = 0.30`

Solving (b): P(B)

We only consider data in the Yes column.

Here:

`(1) = 22`
`(2) = 25` and
`(3) = 30`

`n(B) = (1) + (2) + (3)`

`n(B) = 22 + 25 + 30`

`n(B) = 77`

P(B) is then calculated as:

`P(B) = (n(B))/(Sample)`

`P(B) = (77)/(100)`

`P(B) = 0.77`

Solving (c): P(A n B)

Here, we only consider the similar cell in the yes column and sample 1 row.

i.e. [Supplier 1][Yes]

This is represented as: n(A n B)

`n(A\ n\ B) = 22`

The probability is then calculated as:

`P(A\ n\ B) = (n(A\ n\ B))/(Sample)`

`P(A\ n\ B) = (22)/(100)`

`P(A\ n\ B) = 0.22`

Solving (d): P(A u B)

This is calculated as:

`P(A\ u\ B) = P(A) + P(B) - P(A\ n\ B)`

This gives:

`P(A\ u\ B) = (30)/(100) + (77)/(100) - (22)/(100)`

Take LCM

`P(A\ u\ B) = (30+77-22)/(100)`

`P(A\ u\ B) = (85)/(100)`

`P(A\ u\ B) = 0.85`