A house has two identical sprinkles and humidity sensors that will trigger in: -

Question

A house has two identical sprinkles and humidity sensors that will trigger in:

- 80% of the cases where the humidity is low

-15% of the cases where the humidity is nominal

- 0.5% of the cases where the humidity is high.

The probability of high humidity is 10%, nominal humidity is 50%, and low humidity is 40%.

Describe a Bayesian approach and corresponding queries for computing the following:

1) Probability that the first sprinkler will activate given that the other sprinkler was also activated

2) Probability that the humidity is low given that all both sprinklers are active

3) Probability that the humidity is low given that at least one sprinkle is active

Explanation / Answer

Probability that a sprinkler will activate is computed as:

= Probability of low humidity * Probability that sprinkler would active in case of low humidity + Probability of nominal humidity * Probability that sprinkler would active in case of nominal humidity + Probability of high humidity * Probability that sprinkler would active in case of high humidity

= 0.4*0.8 + 0.5*0.15 + 0.1*0.005 = 0.343

Now, probability that both sprinklers would be activated is computed as:

= 0.4*0.82 + 0.5*0.152 + 0.1*0.0052 = 0.2672525

Now given that the second sprinker is activated, probability that the first sprinkler will also be activated is computed as:

= Probability that both sprinklers are activated / Probability that the second sprinkler will be activated

= 0.2672525 / 0.343

= 0.7792

Therefore 0.7792 is the required probability here.

b) Given that both sprinkers are active, probability that the humidity is low is computed as:

= Probability that humidity is low and both are activated / Probability that both are active

= 0.4*0.82 / 0.2672525

= 0.9579

Therefore 0.9579 is the required probability here.

c) Probability that both the sprinklers are not active is computed as:

= 0.4*(1-0.8)2 + 0.5*(1-0.15)2 + 0.1*(1-0.005)2 = 0.4762525

Given that both sprinkers are not active, probability that the humidity is low is computed as:

= Probability that humidity is low and both are not active / [1 - Probability that both are not active ]

= 0.4*(1-0.8)2 / [1 - 0.4762525 ]

= 0.0305

Therefore 0.0305 is the required probability here

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