To solve this, we'll use the exponential probability distribution, which is used to model the time between events (such as the duration a battery lasts before needing to be recharged). The probability density function (PDF) of the exponential distribution is given by:

$f(x; \lambda) = \lambda e^{-\lambda x}$

Where:

• $x$ is the time (in hours, in this case).
• $\lambda$ is the rate parameter and is equal to $\frac{1}{\mu}$, where $\mu$ is the mean (average) time.

In this problem, the average time is 4.2 hours, so:

$\mu = 4.2 \quad \text{hours} \quad \Rightarrow \quad \lambda = \frac{1}{4.2} \approx 0.2381$

Now, let's determine each probability:

a) Probability that the next charge will last less than 2.3 hours:

The cumulative distribution function (CDF) of the exponential distribution is:

$F(x; \lambda) = 1 - e^{-\lambda x}$

To find the probability that the charge lasts less than 2.3 hours, we compute $F(2.3; \lambda)$:

$P(X < 2.3) = 1 - e^{-0.2381 \times 2.3}$

Let's calculate this.

b) Probability that the next charge will last between 2.3 and 3.3 hours:

The probability for an interval $[a, b]$ is given by:

$P(a \leq X \leq b) = F(b; \lambda) - F(a; \lambda)$

For this part, we calculate $F(3.3; \lambda) - F(2.3; \lambda)$:

$P(2.3 \leq X \leq 3.3) = \left( 1 - e^{-0.2381 \times 3.3} \right) - \left( 1 - e^{-0.2381 \times 2.3} \right)$

c) Probability that the next charge will last more than 2.8 hours:

This is the complement of the CDF:

$P(X > 2.8) = 1 - F(2.8; \lambda) = e^{-0.2381 \times 2.8}$

Let's calculate all these probabilities now.Here are the calculated probabilities:

• a) The probability that the next charge will last less than 2.3 hours is 0.4217.
• b) The probability that the next charge will last between 2.3 and 3.3 hours is 0.1225.
• c) The probability that the next charge will last more than 2.8 hours is 0.5134.

Would you like any additional details or explanations?

Here are 5 related questions to explore:

1. How would the probabilities change if the average battery life increased to 5 hours?
2. What is the cumulative probability that the battery lasts less than 4.2 hours?
3. How do we calculate the expected variance in battery life using the exponential distribution?
4. Can the exponential distribution model be applied to other situations like phone battery life?
5. How does the exponential distribution differ from the normal distribution in this context?

Tip: The exponential distribution is memoryless, meaning the future battery life doesn't depend on how long the battery has already lasted.