Question

Rework problem 27 from section 3.1 of your text, involving a malfunctioning television set. Use the following probabilities instead of those listed in your book: A. A clear picture with probability 0.4 B. Good sound with probability 0.55 C. Good color with probability 0.5 D. Both a clear picture and good sound with probability 0.2 E. Both a clear picture and good color with probability 0.2 F. Both a good sound and good color with probability 0.25 G. A clear picture, good sound, and good color with probability 0.1 What is the probability that the television has exactly one of the three characteristics listed (clear picture, good sound, or good color)?

Answer 1

Let event P be the clear picture

S be the good sound and C be the good color.

P( P and S and C ) = 0.1

P( P and S ) = 0.2

P( P and C ) = 0.2

P( S and C ) = 0.25

So P( P and S and not C ) = 0.2-0.1 = 0.1

P( P and C and not S ) = 0.2-0.1 = 0.1

P( S and C and not P ) = 0.25 - 0.1 = 0.15

P( P) = 0.4

So P( P and not S and not C ) = 0.4-0.1-0.1-0.1 = 0.1

P(S) = 0.55

So P( S and not P and not C ) = 0.55 - 0.1-0.1-0.15 = 0.2

P(C) = 0.5

So P(C and not P and not S ) = 0.5 -0.1-0.1-0.15 = 0.15

From the given information we can draw the following Venn diagram.

Probability that the television has exactly one of the three characteristics = 0.1+0.15+0.2 = 0.45