Consider a virus that is spread by blood transfusion note it

Consider a virus that is spread by blood transfusion (note, it can only spread if the blood was contaminated). Suppose the probability of receiving contaminated blood and then contracting the virus is 0.005, and the probability of being infected if someone receives contaminated blood is 0.20.

(a) Find the probability of blood being contaminated.
(b) Find the probability of not being infected if someone receives contaminated blood.
(c) Find the probability of (not being infected) union (blood is not contaminated).

Solution

(a) Find the probability of blood being contaminated.

Given P(contaminated blood and then contracting the virus) = 0.005

P(infected|contaminated blood)

=P(infected and contaminated blood)/P(contaminated blood) =0.2

So P(contaminated blood) = 0.005/0.2 = 0.025

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(b) Find the probability of not being infected if someone receives contaminated blood.

P(not being infected | contaminated blood)

=P(not being infected and contaminated blood) / P(contaminated blood)

= (P(contaminated blood) - P(infected and contaminated blood)) /P(contaminated blood)

= (0.025-0.005)/0.025

=0.8

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(c) Find the probability of (not being infected) union (blood is not contaminated).

P(infected) = 1-0.025+0.005 = 0.98

P(not being infected U not contaminated)

= P(not being infected) + P(not contaminated) - P(not being infected and not contaminated)

= (0.025-0.005) + (1-0.025) - (1-0.98-0.025+0.005)

=0.995