Suppose that an Ipad has an average lifespan of 5 years. We are interested in the life of the load. What is the probability that an Ipad would last more than 7 years?
The Correct Answer and Explanation is :
To determine the probability that an iPad lasts more than 7 years, assuming an average lifespan of 5 years, we typically model the lifetime of electronic devices using an exponential distribution. The probability density function (PDF) of an exponentially distributed random variable ( T ) with mean ( \lambda ) is:
[
f(t) = \frac{1}{\lambda} e^{-t/\lambda}, \quad t \geq 0
]
where ( \lambda ) is the mean lifespan (in this case, 5 years). The cumulative distribution function (CDF), which gives the probability that the iPad lasts at most ( t ) years, is:
[
P(T \leq t) = 1 – e^{-t/\lambda}
]
We want to find the probability that an iPad lasts more than 7 years, which is given by:
[
P(T > 7) = 1 – P(T \leq 7) = e^{-7/5}
]
Now, calculating:
[
P(T > 7) = e^{-7/5} = e^{-1.4} \approx 0.2466
]
Explanation:
The exponential distribution is commonly used for modeling the lifespan of electronic devices due to its memoryless property, meaning that the probability of survival beyond a certain time is independent of how long the device has already lasted.
In this case, the mean lifespan of an iPad is 5 years. The exponential model assumes that failure happens at a constant rate over time, making it a reasonable approximation for devices that degrade due to wear and tear.
By computing ( P(T > 7) = e^{-7/5} \approx 0.2466 ), we find that there is approximately a 24.66% chance that an iPad lasts more than 7 years. This suggests that while some iPads will last significantly longer than expected, most will fail within or around the 5-year mark.
This model can also be used for reliability analysis of other electronic devices, helping consumers and manufacturers estimate product longevity.