Problem 33 - A First Course in Probability

Question

The number of years a radio functions is exponentially distributed with parameter \(\lambda=\frac{1}{8} .\) If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?

Step-by-Step Solution

Answer

The probability that the used radio will be working after an additional 8 years is approximately \(0.3679\) or \(36.79\%\).

Step 1: Identify the exponential distribution function

The exponential distribution function has the probability density function (pdf) given by: \[f(x) = \lambda e^{-\lambda x},\] where \(x \ge 0\), and the Cumulative Distribution Function (CDF) given by: \[F(x) = 1 - e^{-\lambda x},\] where \(x \ge 0\). In this problem, we have \(\lambda = \frac{1}{8}\).

Step 2: Calculate the probability of the radio working after 8 years

We are interested in finding the probability that the radio will be working after an additional 8 years, i.e., we want to find the probability that it will function for at least 8 years: \[P(X > 8).\] Using the CDF, we can find this probability as: \[P(X > 8) = 1 - P(X \le 8).\] Now, we can use the CDF formula: \[P(X > 8) = 1 - (1 - e^{-\lambda * 8}).\]

Step 3: Substitute the parameter value into the equation

Substitute \(\lambda = \frac{1}{8}\) into the equation: \[P(X > 8) = 1 - (1 - e^{-\frac{1}{8} * 8}).\]

Step 4: Calculate the probability

Now, we compute the value of the probability: \[P(X > 8) = 1 - (1 - e^{-1})\] \[P(X > 8) = 1 - (1 - e^{-1})\] \[P(X > 8) = e^{-1}\]

Step 5: Evaluate the exponential term

Evaluate the exponential term: \[P(X > 8) \approx 0.3679\] The probability that the used radio will be working after an additional 8 years is approximately 0.3679 or 36.79%.