A train arrives at a station and waits 3.5 minutes before departing. Another train arrives at the station 16 minutes later, repeating the cycle. Identify the probability that a train will not be at the station when you arrive. 0.82 0.84 0.22 0.18

To solve this problem, we need to analyze the timing of the trains and the intervals during which they are at the station. 1. **Understanding the Train Schedule:** - The first train arrives and waits for 3.5 minutes. - After it departs, another train arrives 16 minutes later. - Therefore, the total cycle time from the arrival of the first train to the arrival of the second train is \(3.5 + 16 = 19.5\) minutes. 2. **Identifying the Time Intervals:** - During the first 3.5 minutes, the first train is at the station. - After the first train leaves, there are 16 minutes until the next train arrives, during which no train is at the station. 3. **Calculating the Probability:** - The total time in the cycle is 19.5 minutes. - The time when no train is at the station is 16 minutes. - The probability that a train will not be at the station when you arrive can be calculated as the ratio of the time without a train to the total cycle time. \[ P(\text{no train}) = \frac{\text{time without a train}}{\text{total cycle time}} = \frac{16}{19.5} \] Now, let's calculate this probability. \[ P(\text{no train}) = \frac{16}{19.5} \] I'll perform the calculation now. Calculate the value by following steps: - step0: Calculate: \(\frac{16}{19.5}\) - step1: Convert the expressions: \(\frac{16}{\frac{39}{2}}\) - step2: Multiply by the reciprocal: \(16\times \frac{2}{39}\) - step3: Multiply: \(\frac{16\times 2}{39}\) - step4: Multiply: \(\frac{32}{39}\) The probability that a train will not be at the station when you arrive is approximately \(0.8205\), which can also be expressed as \(\frac{32}{39}\). Given the options: - 0.82 - 0.84 - 0.22 - 0.18 The closest answer is **0.82**. Thus, the probability that a train will not be at the station when you arrive is **0.82**.