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Discrete Probability Distributions
Discrete Random Variables
Summary Measures
Continuous Probability Distributions
Statistical Sampling and Regression

PreMBA Analytical Methods
Discrete Probability Distributions: Discrete Random Variables

A random variable has a value that is uncertain and determined by chance events. A discrete random variable will take on values that are separate and distinct. Examples include

  • the number of airplanes owned by an airline
  • the number of people waiting in line at a grocery store checkout
  • the number of cars available for rent by a rental agency
  • the number of patents developed by a research and development department

Imagine selecting one card from a deck of cards. The value of the card you will select is uncertain and determined by chance. Therefore, the value of the card before you choose it is a random variable.

The possible values this random variable can assume are the values of each card in the deck: two, three, four, and so on. Because these values are distinct, indivisible amounts, the random variable is discrete.

After you have selected a card, its value is no longer uncertain and therefore is not random. For example, if you select a five card, five is the value that the discrete random variable has assumed. However, five is not a random variable itself.

Probability Distributions

Each possible value of a random variable has a certain probability associated with it. For example, when you pick a card from a deck, one of the possible values the random variable can assume is five. The probability associated with this value is , or approximately .08, because 4 of the 52 cards are 5s ( simplifies to ).

The probability distribution of a discrete random variable will tell you all of the possible values that variable may take on, as well as the probability associated with each value.

Consider the example of a company that is interested in better understanding the number of orders it receives each week. The discrete random variable in this example is the number of orders per week. The company compiles the number of orders it has received in the past. These values, which range from 41 to 48, are the possible values of the random variable.

The company then determines the future probability of each value based on past data. This probability distribution is illustrated in the table below.

In the distribution P(x = xi) = pi, the following notation is used.

x = the random variable
xi = the ith value of the random variable x
pi = the probability that x will equal xi

Recall from the rules of probability that the probability of an outcome is between 0 and 1 (0 is less than or equal to pi is less than or equal to 1) and that the sum of all probabilities must equal 1, as indicated by the notation below.

The distribution not only tells the company that the number of orders have ranged from 41 to 48 orders per week, but also that most weeks have between 43 and 46 orders. Another way to illustrate this probability distribution is in the form of a graph or visual depiction. A graph for the weekly order volumes is shown here.

This graph shows the probability of each number of weekly orders. The height of each bar displays the probability associated with that number of orders per week. You may notice that in the graph, it is much easier to see the cluster of orders in the range of 43 to 46 orders per week than it was in the table of values.

Binomial Distributions

In the example above, the probability distribution contained a number of possible values that the random variable could assume, ranging from 41 to 48. A binomial distribution is a special form of a discrete probability distribution. The random variable, x, is the number of successes in n independent trials, where each trial can result in only two outcomes—success or failure.

Learn More About the Probability of Success

An experiment or trial with a random variable that obeys a binomial distribution is called a Bernoulli trial. The probability associated with each outcome does not change in repeated trials.

A common example of a random variable that obeys a binomial distribution is the flip of a coin n times, because there are only two possible outcomes—heads or tails. The probability associated with each value does not change in repeated trials. In other words, every time you flip a fair coin, the probability of the outcome being heads is .5 and the probability of the outcome being tails is .5. The outcome of one flip does not affect the probability of the next flip. The random variable is the number of heads (or tails) obtained out of n flips.

If you flip a coin four times (n = 4), what is the probability that the outcome will be heads exactly three times—P(x = 3, n = 4)? To solve this problem, use the following formula, which calculates the probability of x successes in n trials of a variable that obeys a binomial distribution. Note the factorial notation in the formula below, indicated by the (!), as in "n!". If you want to review this concept, click on the link in the right margin.

Read About the Bernoulli Family and its Legacy of Mathematics

P = probability of the outcome of x successes (or failures) in n trials
x = number of successes (or failures)
n = number of trials
p = probability associated with the outcome of success (or failure)

You can use Excel to solve problems that use factorial notation.
Download Factorial Excel Tutorial

In this example, you are interested in the probability of the outcome being heads exactly three times, so you will define success as "heads." The probability of a single toss being heads is .5. A success is defined as a coin flip whose outcome is heads. The number of trials is four, because the coin will be flipped four times. Substituting this information into the formula, you would say that

P = probability of heads exactly three times
x = 3
n = 4
p = .5

Factorial Notation

The probability that four coin flips would result in 3 heads is .25.

Variables that obey a binomial distribution are common in the business world, particularly in quality-control functions.

Consider a cell phone manufacturer with a relatively simple but important quality test at the end of the manufacturing line—each phone is dropped onto a hard surface. If the phone cracks or breaks in any way, it is rejected.

The test is a Bernoulli trial because the result is that the phone is either passed for sale or rejected. If the phone passes, it is considered a success. If it is rejected, it is considered a failure. The probability of a phone breaking and being rejected in a single trial is .01.

What is the probability that five phones will break and be rejected during a shift in which 100 phones are produced? To answer this question, use the formula to calculate the probability of a particular outcome of a variable in a binomial distribution. (This is the same formula used above in the coin flip example.)

P = probability of the outcome
x = number of failures
n = number of trials
p = probability associated with failure

Using the information from the example, you would say that

P = probability of 5 rejects in 100 trials
x = 5
n = 100
p = .01

Substitute these numbers into the formula and solve.

The probability that 5 phones will be rejected during a shift that produces 100 phones is .003.

1. The likelihood of a parts failure in an automobile is a random variable because its value is determined by chance—it is a future unknown associated with a relative probability. Is the number of parts failures that occurred last year a random variable?

Solution 1

2. There are different probabilities associated with each possible value of a random variable. Consider the following table of starting salaries of a group of recent college graduates. Given the probabilities listed in the table below, what is the likelihood that a graduate will earn a starting salary of $83,000?

Solution 2

3. Given the above probability distribution, what is the probability that the starting salary for one of these graduates will be $65,000 or below?

Solution 3

4. What is the probability that a fair coin will land heads up exactly five times in seven flips?

Solution 4

5. Silicon chips are inspected at the completion of the fabrication process. Chips either pass or fail the inspection. The probability of a chip failing is .01. Chips that fail the inspection are destroyed. What is the probability that a production run manufacturing 500 chips will contain 10 failed chips?

Solution 5

You can use Excel to solve for binomial distributions.
Download Binomial Distribution Excel Tutorial
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