Probability distribution

Here you will learn about probability distribution, including theoretical probability, expected frequency, relative frequency and experimental probability.

Students will first learn about probability distribution as part of statistics and probability in 7 th grade.

What is a probability distribution?

A probability distribution describes the complete set of possible outcomes and their associated probabilities for a random variable in an experiment or situation. Two types of probability are theoretical and experimental.

Theoretical probability calculates the likelihood of an event using mathematical principles and formulas, rather than conducting physical experiments or collecting empirical data.

To do this, use the following formula:

For example, the theoretical probability of rolling a 3 on a fair 6 -sided die is \cfrac{1}{6}.

Step-by-step guide: Theoretical probability

Experimental probability is the likelihood of an event calculated by conducting repeated trials or observations and measuring the actual outcomes.

To calculate the experimental probability of an event, you calculate the relative frequency of the event.

To do this, use the following formula:

Step-by-step guide: Experimental probability

Expected frequency represents the predicted number of times an event will occur when an experiment is repeated a specific number of times.

To find the expected frequency, multiply the probability of that event taking place by the number of trials of the experiment, using the following formula:

For example, if a fair 6 -sided die is rolled 12 times, the expected frequency of a 3 is 12\times\cfrac{1}{6}=2.

In 12 rolls of the die, you would expect to roll a 3 twice.

Step-by-step guide: Expected frequency

Relative frequency is the number of times an event happens divided by the total number of outcomes that took place in an experiment, known as the total number of trials or the total number of observations. A frequency, or frequency count, is the number of times a value of the data happens.

To calculate the relative frequency, you can use the formula

For example, when performing an experiment and rolling a 6 -sided die 12 times, the relative frequency could have been 3 (or any number 12 or less).

Step-by-step guide: Relative frequency

Below are some common theoretical probability distributions.

The probability distribution of rolling a fair 6 -sided die is:

The probability distribution of flipping a fair coin is:

The probability distribution of picking a card from a deck is:

The probability distribution of spinning a color on a roulette wheel is:

What is a probability distribution?

Common Core State Standards

How does this relate to 7 th grade math?

• Grade 7 – Statistics and Probability (7.SP.C.5)
  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around \cfrac{1}{2} indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

• Grade 7 – Statistics and Probability (7.SP.C.7)
  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

• Grade 7 – Statistics and Probability (7.SP.C.8)
  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

How to use probability distribution

There are a lot of ways to use probability distribution. For more specific step-by-step guides, check out the probability pages linked in the “What is a probability distribution?” section above or read through the examples below.

Probability distribution examples

Example 1: selecting a random letter

Lettered cards spell out the word mathematics.

The cards are mixed up and one is selected at random.

What is the probability that the card selected is a consonant? Display it on the probability scale.

1. Determine all possible outcomes.

There are 11 cards altogether in the word MATHEMATICS.

2Determine the frequency of the event occurring.

The set of consonants are all letters except: A, \, E, \, I, \, O, and U. There are 7 consonant cards in the word MATHEMATICS.

3Substitute these values into the formula. Write your answer as a fraction, decimal or percentage.

The frequency of the event occurring is 7 and the total frequency of possible outcomes is 11, the theoretical probability is

Example 2: finding experimental probability relative frequency

A 3 -sided spinner numbered 1, \, 2, and 3 is spun and the results are recorded.

Find the probability distribution for the 3 -sided spinner from these experimental results.

Draw a table showing the frequency of each outcome in the experiment.

A table of results has already been provided. You can add an extra column for the relative frequencies.

Write the experimental probability (relative frequency) of the required outcome(s).

Divide each frequency by 110 to find the relative frequencies.

Example 3: calculating relative frequency or a discrete random variable

A coin lands on heads 18 times, having been flipped 40 times. Calculate the relative frequency of the coin toss showing tails.

Example 4: calculating the number of trials if relative frequency is known

An experiment rolling a fair 6 -sided dice landed on the number 4 on 35 occasions.

The relative frequency of rolling a 4 was calculated as 0.14. Find the number of times the dice was rolled.

Example 5: calculating expected frequency

The probability that John will win a game of tennis is 0.75. If John plays 80 games of tennis, how many can he expect to win?

Example 6: exponentially small number of successes

There are 26 letters in the English alphabet. A statistician randomly chooses 3 letters. The probability that the letters ‘pmf’ or ‘cdf’ are chosen is 0.000128. If the statistician does 5,000 trials, how many times can she expect to choose ‘pmf’ or ‘cdf’?

Teaching tips for probability distribution

• When introducing this topic, start with the probability of an event (preferably independent events) that has a small number of total possible values and a small number of occurrences, so that students can spend their time understanding probability, instead of calculating.

• To extend students’ knowledge, apply this topic to real world data science topics, including large sample sizes or having students interpret a probability dataset shown in a histogram.

Easy mistakes to make

• Assuming that all probability distributions are the same
  It is possible for students to assume that all the probabilities in a distribution will be the same. Encourage students to focus on only the theorems they are taught, but also be clear that as they continue to learn about probability, they will learn about new distributions.

• Not knowing the meaning of the words “and“ and “or” in probability questions
  It is important that students understand the difference between “and” and “or” and when they need to multiply probabilities and when they should add probabilities. Typically, the word “or” means to add the probabilities. “And” will involve multiplication.

• Confusing parameters and statistics
  Remind students that parameters describe the population and they are fixed. While statistics describe a sample of the population and therefore can vary.

The next lessons are

• Units of measurement
• Represent and interpret data
• Ratio

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