Question

Between \(12: 00 \mathrm{PM}\) and \(1: 00 \mathrm{PM}\) cars arrive at Citibank's drive-thru at the rate of 6 cars per hour \((0.1\) car per minute). The following formula from probability can be used to determine the probability that a car will arrive within \(t\) minutes of \(12: 00 \mathrm{PM}\).$$F(t)=1-e^{-0.1 t}$$ (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 \(80 \%\) (c) Is it possible for the probability to equal \(100 \%\) ? Explain.

Short answer

(a) 6.93 minutes, (b) 16.09 minutes, (c) No, it can't reach exactly 100%.

Step-by-step solution

Understand the Formula

The given formula is \[F(t) = 1 - e^{-0.1t}\] which represents the probability that a car will arrive within t minutes.

Calculate for 50% Probability

To find the time it takes for the probability to reach 50%, set \[F(t) = 0.5\] and solve for t.\[0.5 = 1 - e^{-0.1t}\]Rearrange to solve for the exponent term:\[e^{-0.1t} = 0.5\]Take the natural logarithm (ln) of both sides:\[ -0.1t = \text{ln}(0.5)\]Solve for t:\[t = -\frac{\text{ln}(0.5)}{0.1} \approx 6.93 \text{ minutes}\]

Calculate for 80% Probability

To find the time it takes for the probability to reach 80%, set \[F(t) = 0.8\] and solve for t.\[0.8 = 1 - e^{-0.1t}\]Rearrange to solve for the exponent term:\[e^{-0.1t} = 0.2\]Take the natural logarithm (ln) of both sides:\[ -0.1t = \text{ln}(0.2)\]Solve for t:\[t = -\frac{\text{ln}(0.2)}{0.1} \approx 16.09 \text{ minutes}\]

Determine if Probability Can Reach 100%

The probability formula given is \[F(t) = 1 - e^{-0.1t}\]As t approaches infinity, \[e^{-0.1t} \text{ approaches } 0\]Thus, \[F(t) \text{ approaches } 1 \text{ but never fully equals } 1\]Therefore, the probability can never reach exactly 100%.

Key concepts

Exponential Distribution

An exponential distribution is a type of continuous probability distribution. It is often used to model the time between events in a process where events occur continuously and independently at a constant average rate. This makes it useful for scenarios such as calculating the time until a car arrives at a drive-thru. In our example, cars arrive at a rate of 6 per hour, which corresponds to an average rate (\textlambda) of 0.1 cars per minute.

The exponential distribution has a probability density function given by: \[ f(t) = \textlambda e^{-\textlambda t} \]
Here, \textlambda represents the rate parameter. Note that higher values of \textlambda mean the events happen more frequently. The cumulative distribution function (CDF) is another essential concept, and it's used to determine the probability that a car will arrive within a certain time, given by: \[ F(t) = 1 - e^{-\textlambda t} \]

Natural Logarithm

The natural logarithm (ln) is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. In probability and statistics, the natural logarithm often helps solve equations involving exponential functions.

In our exercise, we use ln to find the time t for which the probability function F(t) reaches certain values like 50% or 80%. For instance, when we want F(t) to be 0.5, we set up the equation: \[ e^{-0.1t} = 0.5 \]
To solve for t, we take the natural logarithm of both sides, as it transforms the exponential equation into a linear one: \[ -0.1t = \text{ln}(0.5) \]
This can be further simplified and solved to determine the time needed.

Probability Calculation

Calculating probabilities in the context of exponential distributions involves using the CDF formula \[ F(t) = 1 - e^{-0.1t} \] with given values. For example, to determine how long it takes for the probability to reach 50%, 80%, or any other percentage, you set F(t) to the desired probability value.

For 50%, the calculation steps are:

• Set F(t) to 0.5: \[ 0.5 = 1 - e^{-0.1t} \]
• Rearrange: \[ e^{-0.1t} = 0.5 \]
• Take the natural log: \[ -0.1t = \text{ln}(0.5) \]
• Solve for t: \[ t = -\frac{\text{ln}(0.5)}{0.1} \]
• Approximate: \[ t \text{ ≈ 6.93 minutes} \]

For 80%, follow similar steps and you get approximately 16.09 minutes. This method shows how to interpret and work with exponential distributions practically.

Continuous Probability Distribution

A continuous probability distribution describes the likelihood of all possible values of a continuous random variable. Unlike discrete distributions, where outcomes are countable, continuous distributions cover a continuous range of values.

The exponential distribution is one such continuous probability distribution. The key feature is that it models the time between independent events that happen at a constant average rate. For example, with a drive-thru where cars arrive continuously.
The probability density function (pdf) and cumulative distribution function (CDF) are crucial. The CDF gives the probability that the time until the next event is less than or equal to a specified value. This is represented by: \[ F(t) = 1 - e^{-\textlambda t} \] As time t increases, the exponential term becomes very small, and the CDF approaches 1 but never fully reaches it. This is why the probability can't equal 100%. This result underscores that while you can expect an event almost certainly after a sufficiently long time, there's always a tiny chance it hasn't occurred yet.