Problem 24 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

TIME MANAGEMENT A bakery turns out a fresh batch of chocolate chip cookies every 45 minutes. Tina arrives (at random) at the bakery, hoping to buy a fresh cookie. Use an appropriate uniform density function to find the probability that Tina arrives within 5 minutes (before or after) the time that the cookies come out of the oven.

Step-by-Step Solution

Answer

The probability is \( \frac{2}{9} \).

Step 1 - Define the Time Interval

Each fresh batch of cookies is made every 45 minutes, so consider an interval from 0 to 45 minutes (one baking cycle).

Step 2 - Set Up the Uniform Density Function

The arrival time of Tina can be modeled using a uniform density function since she arrives randomly over the 45 minutes. This means the probability density function (PDF) is constant across the interval.

Step 3 - Determine the PDF Expression

For a uniform distribution, the PDF is given by \( f(t) = \frac{1}{b - a} \) where \( a \) and \( b \) are the limits of the interval. Here, \( a = 0 \) and \( b = 45 \). So, \( f(t) = \frac{1}{45} \).

Step 4 - Define the Desired Time Window

Tina needs to arrive within 5 minutes before or after the time when the cookies come out. This creates a 10-minute interval (5 minutes before plus 5 minutes after).

Step 5 - Calculate the Probability

The probability that Tina arrives in this 10-minute window is the proportion of this interval within the 45-minute period. Calculate this using \( P = f(t) \times (10) = \frac{10}{45} = \frac{2}{9} \).

Key Concepts

Probability Density Function

A Probability Density Function (PDF) allows us to see how the probabilities are distributed over an interval. For continuous random variables, the PDF gives us a way to calculate the probability that a variable falls within a particular range.
When dealing with a PDF, it is important to remember that the area under the curve within a given range represents the probability of a variable falling within that range. Unlike with discrete distributions, where you're summing probabilities, here you're integrating the PDF over the interval of interest.
In the case of our bakery example, Tina’s arrival time is treated as a continuous random variable spread uniformly over the 45-minute baking cycle. Hence, the PDF is represented uniformly across the interval, meaning Tina has an equal chance of arriving at any time from 0 to 45 minutes.

Uniform Distribution

A Uniform Distribution is a type of probability distribution where all outcomes are equally likely within a defined interval. It is called 'uniform' because the probability remains consistent across the interval.
In our exercise, the interval is from 0 to 45 minutes, representing the baking cycle. We know the function is uniform because Tina's arrival is equally likely at any minute within this interval.
For a uniform distribution, the probability density function (PDF) is constant and calculated by the formula: \( f(t) = \frac{1}{b - a} \) where \( a \) and \( b \) are the limits of the interval. By applying this to our case, \( f(t) = \frac{1}{45} \). This shows that Tina has an equal chance of arriving at any time.

Time Management

Time management is crucial when dealing with probabilities involving time intervals. By dividing tasks into intervals, you can assess the likelihood of events occurring within those specific time frames.
For example, in the bakery problem, understanding the uniform distribution helps Tina manage her expectations better. She would know that the chances of arriving exactly when a new batch is ready are low unless she arrives randomly within a short time window.
Effective time management would involve creating strategies such as timing her visit closer to when cookies are expected to be fresh, thus maximizing her chance of enjoying freshly baked cookies. Breaking down her timing into manageable intervals also improves her decision-making process, allowing Tina to plan her day with a realistic outlook on probabilities.