Question 1193223: At a restaurant, the time a customer has to wait before being seated follows an exponential distribution and is given by the probability density function (pdf):
f(t)=1e−1t
Find the probability that a customer will have to wait at least 4 minutes for a table.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! **1. Identify the Distribution**
* The given probability density function (PDF)
f(t) = 1/e * e^(-t)
represents an **exponential distribution** with a rate parameter (λ) of 1.
**2. Find the Probability of Waiting at Least 4 Minutes**
* We need to find P(T ≥ 4), where T is the waiting time.
* **For an exponential distribution, the cumulative distribution function (CDF) is given by:**
F(t) = 1 - e^(-λt)
* **To find P(T ≥ 4), we can use the complementary probability:**
P(T ≥ 4) = 1 - P(T < 4)
P(T ≥ 4) = 1 - F(4)
* **Calculate F(4):**
F(4) = 1 - e^(-1 * 4)
F(4) = 1 - e^(-4)
F(4) ≈ 1 - 0.0183
F(4) ≈ 0.9817
* **Calculate P(T ≥ 4):**
P(T ≥ 4) = 1 - F(4)
P(T ≥ 4) = 1 - 0.9817
P(T ≥ 4) ≈ 0.0183
**Therefore, the probability that a customer will have to wait at least 4 minutes for a table is approximately 0.0183 (or 1.83%).**
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