the probability that a key component of a stand -by generator will fail during a power outage is q. A generator works if at least half of its key component work.Assuming that the the generator has four key components. find the probability that the generator will work
All of the parts have to work.
probability of part failing =q
probability of part working = (1-q)
4 parts, binomial distribution
4Ck = 4!/[k!(4-k)!]
4C4 = 1 , 4C3= 4 , 4C2= 6
P4 of 4 working = 1 (1-q)^4 q^0= (1-q)^4
P3 of 3 working = 4 (1-q)^3 q^1
P2 of 2 working = 6 (1-q)^2 2^2
add those
To find the probability that the generator will work, we need to consider the probability that at least half of its key components will work.
Let's calculate the probability of each possible scenario:
1. All four key components work: In this case, the generator will definitely work. The probability of this happening is q^4.
2. Three key components work: In this case, the generator will work since at least half of its components are functioning. There are four possible ways this can happen (by choosing one component to fail), and each possibility has a probability of q^3. So, the probability is 4*q^3.
3. Two key components work: In this case, the generator will still work since two out of four components is more than half. There are six ways this can happen (by choosing two components to fail), and each possibility has a probability of q^2. So, the probability is 6*q^2.
Finally, we add up all these probabilities to get the total probability that the generator will work:
Total probability = q^4 + 4*q^3 + 6*q^2