Solve the problem.
-Between 7:00 AM and 8:00 AM, trains arrive at a subway station at a rate of 15 trains per hour (0.25 trains per minute). The following formula from statistics can be used to determine the probability that a train will arrive within t
Minutes of 7:00 AM. $$\mathrm { F } ( \mathrm { t } ) = 1 - \mathrm { e } ^ { - 0.25 \mathrm { t } }$$ Determine how many minutes are needed for the probability to reach 20%.
A) 0.89 min
B) 2.35 min
C) 5.34 min
D) 4.12 min

Question

Quizplus · Accepted Answer

To find the time $t$ when the probability $F(t)$ reaches 20% (or 0.20), we set the given formula equal to 0.20 and solve for $t$:$$0.20 = 1 - e^{-0.25t}$$Solving for $t$, we first subtract 1 from both sides and multiply by -1:$$e^{-0.25t} = 0.80$$Taking the natural logarithm (ln) of both sides to solve for $t$:$$-0.25t = \ln(0.80)$$$$t = \frac{\ln(0.80)}{-0.25}$$$$t \approx 0.89$$Thus, it takes approximately 0.89 minutes for the probability that a train will arrive to reach 20%.

Solve the Problem F(t)=1−e−0.25t\mathrm { F } ( \mathrm { t } ) = 1 - \mathrm { e } ^ { - 0.25 \mathrm { t } }F(t)=1−e−0.25t

Solve the Problem $\mathrm { F } ( \mathrm { t } ) = 1 - \mathrm { e } ^ { - 0.25 \mathrm { t } }$