Suppose that 50 identical batteries are being tested. After 8 hours of continuous use, assume that a given battery is still operating with a probability of 0.70 and has failed with a probability of 0.30.
What is the probability that fewer than 40 batteries will last at least 8 hours?
.0598
To answer this question, we need to calculate the cumulative probability from 0 to 39 batteries that will last at least 8 hours.
We can use the binomial distribution to calculate the probability of fewer than 40 batteries lasting at least 8 hours. The binomial distribution is used to calculate the probability of a certain number of successes in a fixed number of independent Bernoulli trials (where each trial has two possible outcomes: success or failure).
In this case, the number of trials is 50 (the number of batteries), the probability of success is 0.70 (the probability that a battery lasts at least 8 hours), and we want to find the probability of fewer than 40 successes (batteries lasting at least 8 hours).
Using a statistical software, a spreadsheet program, or a calculator with a binomial distribution function, you can find the cumulative probability directly.
For example, in Excel, you can use the BINOM.DIST function with the parameters:
- number of successes = 39 (because we want fewer than 40 batteries to last at least 8 hours)
- number of trials = 50
- probability of success = 0.70
- cumulative = TRUE (to get the cumulative probability)
In Excel, the formula would be:
=BINOM.DIST(39, 50, 0.70, TRUE)
This will give you the probability that fewer than 40 batteries will last at least 8 hours.
Alternatively, you can use statistical software like R or Python to calculate the cumulative probability using the binomial distribution function.
Keep in mind that the exact calculation may vary depending on the software or calculator you are using, but the concept remains the same.