A manufacturer has observed that a certain product tends to fail either very early in its life span or quite late. The engineers are interested in studying a particular product going into cellular phones. They know that the average life span of the product is 1,600 days. What is the probability that the product fails in the first 100 days or after 2,400 days?
If there really is a tendency for the product to fail early or not at all (or very much later), then there is no way of predicting this from the information you have been given. What you CAN predict is the probability of failure in a specific time interval if the probability of failure is uniform at all times. You would use an exponential formula
dN/dt = - kN for the number N that had not failed, and get k from the mean life, much as in a radioactive decay calculatikon
To calculate the probability that the product fails in the first 100 days or after 2,400 days, we can use the exponential distribution.
First, let's calculate the failure rate (k) using the average life span of the product, which is 1,600 days. The failure rate is given by k = 1 / mean life span.
k = 1 / 1,600 = 0.000625 per day.
Now, to calculate the probability of failure within a specific time interval, we can use the exponential distribution formula. The formula is:
P(t) = 1 - e^(-kt)
Where P(t) is the probability of failure within time t, e is the base of the natural logarithm (approximately 2.71828), and k is the failure rate.
Let's calculate each probability separately:
1. Probability of failure in the first 100 days:
P(100) = 1 - e^(-0.000625 * 100) ≈ 0.0625 (or 6.25%)
2. Probability of failure after 2,400 days:
P(2,400) = 1 - e^(-0.000625 * 2,400) ≈ 0.8467 (or 84.67%)
So, the probability that the product fails in the first 100 days or after 2,400 days is approximately 6.25% + 84.67% ≈ 90.92%.