Probability of crossing a point in a given time window

The probability of a car passing a certain intersection in a 20 minute windows is 0.9. What is the probability of a car passing the intersection in a 5 minute window? (Assuming a constant probability throughout)

Solution 1:

Define events: \begin{eqnarray*} C_{20} &=& \text{"A car passes in a 20-minute window"} \\ C_5 &=& \text{"A car passes in a 5-minute window"}. \end{eqnarray*}

Then, \begin{eqnarray*} 0.9 &=& P(C_{20}) \\ &=& 1-P(C_{20}^c) \\ &=& 1-P(C_5^c)^4 \qquad\text{(assuming independence of cars passing)} \\ &=& 1-(1-P(C_5))^4 \\ \therefore (1-P(C_5))^4 &=& 0.1 \\ 1-P(C_5) &=& (0.1)^{1/4} \\ P(C_5) &=& 1-(0.1)^{1/4} \\ &\approx& 0.438. \end{eqnarray*}