Cars on a Road

On a certain road the probability of seeing at least one car pass during a 20-minute window is $\tfrac{609}{625}$ . Cars arrive at a constant rate, and counts over disjoint intervals are independent. What is the probability of seeing at least one car during a 5-minute window?

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#Independence over blocks

Let $p$ be the probability of no car in a 5-minute block. A 20-minute window is four disjoint 5-minute blocks, and independence multiplies their no-car chances,

\PP(\text{no car in 20 min}) = p^4. \tag{1}

#Take the fourth root

The no-car chance over 20 minutes is the complement of the given figure,

p^4 = 1 - \frac{609}{625} = \frac{16}{625} = \left(\frac{2}{5}\right)^4 \quad\Longrightarrow\quad p = \frac{2}{5}. \tag{2}

Each five-minute block independently shows no car with probability two fifths. Four independent blocks multiply, so the silent twenty minutes carries the fourth power, and a single block sees at least one car with probability three fifths.

#Read it off

\PP(\text{at least one car in 5 min}) = 1 - \frac{2}{5} = \frac{3}{5}. \tag{3}

The long-window certainty hides a fairly sparse road. A car shows up in any given five minutes only three times in five, but stacking four of those windows pushes the chance of total silence down to $\tfrac{16}{625}$ .