Probability that an electrical switch which is kept in dryne

Probability that an electrical switch, which is kept in dryness, fails during the guarantee period, is 1%.
If the switch is humid, the failure probability is 8%. Assume that 90% of switches are kept in dry conditions,
whereas remaining 10% are kept in humid conditions.

(a) What is the probability that the switch fails during the guarantee period?

(b) If the switch failed during the guarantee period, what is the probability that it was kept in humid
conditions?

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Solution

Solution:

Let F be for failure, H for Humid and D for Dry.

Its given that probability of failure during dryness is 1 % i.e. P(F/D) = 0.01 also given that probaility of failure during humid is 8 %. i.e. P(F/H)=0.08

Further it is also mention P(D)=0.9 and P(H)=0.1

(a) So the probability that the switch fails during the guarantee period is P (F) = P(F/D)P(D)+P(F/H)P(H)=0.01*0.9+0.08*0.01=0.009+0.008=0.017

(b) If the switch failed during the guarantee period, probability that it was kept in humid conditions is P(H/F) = P(HF)/P(F) = P(F/H)*P(H)/P(F) = 0.08*0.1/0.017=0.4706