Question

# A system with five components, which functions if at least three of the components function, is...

A system with five components, which functions if at least three of the components function, is designed to work outdoors. Suppose that the components function with probability 0.9 if the temperature is above freezing, and they function with probability 0.7 if the temperature is below freezing. The weather forecast for tomorrow predicts there is a probability of 0.40 that the temperature will be below freezing. What is the probability that the system functions tomorrow?

We have two binomial distributions here. Say X be the number of components which function.

When the temperature is above freezing, Probability = 0.6, X~ B(0.9, 5)
When the temperature is below freezing, Probability = 0.4, X~ B(0.7, 5)

We need to find P(X>=3) for both the distributions such that the system will function tomorrow.

P(System functions tomorrow) = P(Above Freezing)*P(X>=3 | Above Freezing) + P(Below Freezing)*P(X>=3 | Below Freezing)

For simplification: AF = Above freezing, BF = Below freezing
We know that,

P(X>=3 | AF) = P(X=3 | AF) + P(X=4 | AF) + P(X=5 | AF)

P(X>=3 | AF) = 0.99144

Similarly, P(X>=3 | BF) = 0.83692

Hence, Probability = 0.6*0.99144 + 0.4*0.83692 = 0.929