Probability Between 5: 00 PM and 6: 00 PM, cars arrive at...

Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within $t$ minutes of 5: 00 PM. $$F(t)=1-e^{-0.15 t}$$ (a) Determine how many minutes are needed for the probability to reach $50 \%$. (b) Determine how many minutes are needed for the probability to reach $80 \%$.

Probability Between 5: 00 PM and 6: 00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour $(0.15$ car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within $t$ minutes of 5: 00 PM.
$$F(t)=1-e^{-0.15 t}$$
(a) Determine how many minutes are needed for the probability to reach $50 \%$.
(b) Determine how many minutes are needed for the probability to reach $80 \%$.

Key Concepts

Solving Exponential Equations Using Logarithms

When determining the time required for the probability of an event to reach a certain level, one often needs to solve equations involving the exponential function. This typically involves isolating the exponential term and applying the natural logarithm to both sides of the equation to solve for the variable of interest.

Rate Parameter

The rate parameter, often denoted by lambda (?), defines the expected number of events per unit time in a Poisson process. It is central to the exponential distribution as it determines the scale of the distribution, affecting how quickly the cumulative probability increases over time.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) represents the probability that a random variable takes on a value less than or equal to a certain point. In the context of the exponential distribution, the CDF is used to determine the probability that an event, such as an arrival, will occur within a specified time period.

Exponential Distribution

The exponential distribution is a continuous probability distribution that models the time between events in a Poisson process. It is characterized by a constant rate at which events occur, meaning the probability of an event occurring in a given time interval is constant regardless of the time elapsed.

Transcript

Please give Ace some feedback