Final answer:
The question asks for the probability of 40 lightbulbs, each with a mean life of 2 months, lasting at least 7 years. The solution involves understanding the exponential distribution, but due to the complexity, a simulation or advanced probability techniques might be required to accurately calculate such a low probability.
Step-by-step explanation:
The question pertains to the probability that 40 lightbulbs, each with a mean life of 2 months, will last at least 7 years. To solve this, we need to understand the exponential distribution, which often describes the time until failure for electronic components like lightbulbs. The mean lifetime (μ) is given as 2 months, and the total time we're interested in is 7 years, or 84 months.
We first determine how many bulbs we would expect to use over 7 years:
84 months ÷ 2 months per bulb = 42 bulbs. However, we are interested in the probability of using 40 or fewer bulbs. Since individual bulb lifetimes are independent, and we expect to use 42 bulbs in 7 years, the probability of needing at most 40 bulbs is fairly low. This kind of problem would typically be approached using a Poisson or a binomial distribution, but these are not valid approaches here due to the large number of expected events (bulb replacements) and the very low probability desired. Therefore, this problem would either require a complex simulation or could be approximated using advanced probability techniques that consider the process is memoryless.
Typically, the answer would be calculated using the formula P(T < t) = 1 - e-t/μ, where T is the random variable representing the lifespan of the bulb. However, in this instance, we are looking at 40 bulbs, and so the cumulative calculation is not so straightforward, and in practice, a simulation or numeric calculation approach would be needed. The exact answer would require further calculation with appropriate software or statistical tables.