Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A bank has two automatic tellers at its main office and two at each of its three branches. The number of machines that break down on a given day, along with the corresponding probabilities, are shown in the following table. $$\begin{array}{l}\text { Machines That } \\\\\text { Break Down } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & .43 & .19 & .12 & .09 & .04 \\ \hline\end{array}$$ $$\begin{array}{lcccc}\hline \text { Machines That } & & & & \\ \text { Break Down } & 5 & 6 & 7 & 8 \\\\\hline \text { Probability } & .03 & .03 & .02 & .05 \\\\\hline\end{array}$$ Find the expected number of machines that will break down on a given day.

Step-by-Step Solution

Answer

The expected number of machines that will break down on a given day is 1.93, meaning that almost two machines are expected to break down daily across the main office and its three branches.

Step 1: List the given probabilities and possible outcomes

The table provided gives the number of machines breaking down and their corresponding probabilities: 0 machines, probability: 0.43 1 machine, probability: 0.19 2 machines, probability: 0.12 3 machines, probability: 0.09 4 machines, probability: 0.04 5 machines, probability: 0.03 6 machines, probability: 0.03 7 machines, probability: 0.02 8 machines, probability: 0.05

Step 2: Calculate the expected value for the number of machines breaking down

Now, we will calculate the expected value using the formula mentioned in the analysis. Multiply each possible outcome (number of machines breaking down) with its corresponding probability and then sum the results. Expected value = (0 * 0.43) + (1 * 0.19) + (2 * 0.12) + (3 * 0.09) + (4 * 0.04) + (5 * 0.03) + (6 * 0.03) + (7 * 0.02) + (8 * 0.05) Expected value = 0 + 0.19 + 0.24 + 0.27 + 0.16 + 0.15 + 0.18 + 0.14 + 0.40 Expected value = 1.93

Step 3: Interpret the result

The expected number of machines that will break down on a given day is 1.93. This indicates that, on average, almost two machines are expected to break down daily across the main office and its three branches.

Key Concepts

Probability Distribution

A probability distribution is a fundamental concept in probability theory that shows how probabilities are spread across various outcomes in a particular experiment. In our exercise, the probability distribution provides the likelihood of a certain number of teller machines breaking down on any given day. To understand probability distributions, consider the following key points:

• Each outcome (0, 1, 2, ... 8) represents a possible number of machines that could break down.
• Each outcome has a corresponding probability, which indicates the chance of that outcome occurring.

For example, the probability of no machines breaking down is 0.43, while the probability of all eight machines breaking down is 0.05.

These probabilities must sum up to 1, as this ensures that the function represents a complete set of potential outcomes. By examining the distribution, we get an insight into how likely each scenario is, thus providing a full picture of the situation we might expect on any given day.

Random Variables

A random variable is a key concept in statistics and probability theory that associates numerical values with each possible outcome of an experiment. In the scenario from the exercise, the random variable is the number of machines breaking down on a given day.

This particular random variable is discrete, meaning it can take on specific values within a finite range, which in our case are whole numbers from 0 to 8.

• The role of the random variable is to bridge the event of machines breaking down with mathematical probabilities.
• The expected value of a random variable provides a single, summary statistic of its distribution.

By calculating the expected value, we find the average outcome we anticipate over many trials or days. This is a powerful tool as it helps in making predictions and preparing for future occurrences. In our case, the expected value indicates that we should typically plan for around 2 machines breaking down.

Probability Theory

Probability theory is the branch of mathematics that deals with analyzing random phenomena and quantifying uncertain events. It forms the basis for all calculations involving probabilities, like expected values and distributions.

Some key aspects of probability theory evident in the exercise include:

• Outcomes: The distinct scenarios (0 through 8 machines breaking down) that can result from a random process.
• Probabilities: Numerical values representing the likelihood of each outcome occurring.
• Expected Value: A measure that gives us an average of all possible outcomes, weighted by their probabilities.

Probability theory allows us to distill complex scenarios into manageable insights. The expected value, derived through this theory, guides us to make informed decisions by predicting tendencies within the random events. In our example, understanding probability theory helps us manage the bank's expectations regarding machinery failure.