SOLUTION: The chance that the front light on your bike will fail is 0.2, the chance that your rear light will fail is 0.1 and the chance that both will fail is 0.04. What is the chance that

Question 1019046: The chance that the front light on your bike will fail is 0.2, the chance that your rear light will fail is 0.1 and the chance that both will fail is 0.04. What is the chance that both lights will work? (regardless of the answer you should do something about this situation of course).
Give your answer as a proportion, rounding to two decimal places.
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Question:
The chance that the front light on your bike will fail is 0.2, the chance that your rear light will fail is 0.1 and the chance that both will fail is 0.04. What is the chance that both lights will work? (regardless of the answer you should do something about this situation of course).
Give your answer as a proportion, rounding to two decimal places.

Solution:
Define events:
F=event that front light fails
R=event that the rear light fails
Given P(F)=0.2, P(R)=0.1, and P(F∩R)=0.04
From statistics, we know that
P(F∪R)=P(F)*P(R)-P(F∩R)
=0.2+0.1-0.04
=0.26 (probability that either light (or both) will fail).......(1)
We require the probability that both lights work, or both lights do not fail.
This is expressed as P(~F∩~R) which by De Morgan's law is equivalent to
P(~(F∪R)
=1-P(F∪R)
=1-0.26 (using (1) above)
=0.74
Therefore, the probability that both lights will work is 0.74.

This problem can also be solved quite easily using a Venn Diagram.

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