The time till failure of a particular TV brand has an exponential distribution. Data shows that 12% of TV's have failed by 1,600 hours. What is the probability that TV's will still be working after 7,100 hours

The time till failure for an exponentially distributed random variable can be modeled by the probability density function (PDF):

\[
f(t) = \lambda e^{-\lambda t}
\]

where \( \lambda \) is the rate parameter, and \( t \) represents time. The cumulative distribution function (CDF) for the exponential distribution, which gives the probability that the time till failure is less than or equal to some value \( t \), is:

\[
F(t) = 1 - e^{-\lambda t}
\]

Given the problem:

- 12% of TVs fail by 1,600 hours, which translates to \( F(1600) = 0.12 \).
- We need to find the probability that TVs will still be working after 7,100 hours, i.e., \( P(T > 7100) \).

### Step 1: Determine the rate parameter (\( \lambda \))

We know from the CDF that:

\[
F(1600) = 1 - e^{-\lambda \cdot 1600}
\]

Given \( F(1600) = 0.12 \), we can solve for \( \lambda \):

\[
0.12 = 1 - e^{-\lambda \cdot 1600}
\]
\[
e^{-\lambda \cdot 1600} = 0.88
\]
\[
-\lambda \cdot 1600 = \ln(0.88)
\]
\[
\lambda = -\frac{\ln(0.88)}{1600}
\]

### Step 2: Calculate the probability that the TV lasts longer than 7,100 hours

The probability that the TV will still be working after 7,100 hours is given by:

\[
P(T > 7100) = 1 - F(7100) = e^{-\lambda \cdot 7100}
\]

Let's calculate \( \lambda \) and the final probability. I'll do the math now.The rate parameter (\( \lambda \)) for the exponential distribution is approximately \( 7.99 \times 10^{-5} \).

The probability that a TV will still be working after 7,100 hours is approximately **0.567**, or 56.7%.

Would you like more details on any step or further clarifications?

Here are 5 related questions you might find interesting:
1. How is the exponential distribution applied in other real-world failure models?
2. What is the expected time until failure based on the calculated \( \lambda \)?
3. How do other distributions, like the Weibull distribution, compare to the exponential distribution in modeling failure?
4. Can we calculate the probability that a TV fails between two given time intervals?
5. How does increasing the rate parameter \( \lambda \) affect the lifespan of devices?

**Tip**: In exponential distributions, the expected value (mean) is the inverse of the rate parameter \( \lambda \), which gives a quick estimate of the average time to failure.

The time till failure of a particular TV brand has an exponential distribution. Data shows that 12% of TV's have failed by 1,600 hours. What is the probability that TV's will still be working after 7,100 hours?

Discover how to calculate the probability that a TV brand will still be working after 7,100 hours, given that 12% of TVs fail by 1,600 hours using the exponential distribution. This problem explores key probability concepts like the exponential distribution, the probability density function, and more. Suitable for high school students in Grades 11-12.

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