Final answer:
To answer the student's question, we use the binomial probability formula to calculate the probability of fewer than two bulb failures, exactly no failures, and more than four failures, considering that the failure of each bulb is independent with a probability of failure of 0.26.
Step-by-step explanation:
To calculate the probability of various outcomes for the street lights with new bulbs, we use the binomial probability formula, which is appropriate since each bulb has only two possible outcomes (failure or no failure) and each bulb fails independently.
To find the probability that fewer than two of the original bulbs fail within 50000 hours of operation, we consider two scenarios: no bulbs failing and one bulb failing. The binomial probability formula is:
P(X = x) = (nCx) * (p^x) * (1-p)^(n-x)
Where:
- n = total number of trials
- X = number of successes (in our case, failures)
- p = probability of success on a single trial
- nCx = combination of n items taken x at a time
With n = 12 and p = 0.26, the probability that exactly zero bulbs fail (X = 0) is calculated using the formula, as is the probability that exactly one bulb fails (X = 1). The two probabilities are then added together to give us the total probability of fewer than two failures.
The probability that no bulbs will have to be replaced within 50000 hours of operation (X = 0) is a special case of the previous calculation with the same formula.
To find the probability that more than four of the original bulbs will need replacing within 50000 hours, we need to calculate the complement of the probability that four or fewer bulbs fail. This can be done by summing the probabilities of 0, 1, 2, 3, and 4 bulbs failing and subtracting that sum from 1.