4. A bag contains $ 3 $ green counters, $ 5 $ white counters and $ 4 $ red counters. A counter is selected at random. Calculate the probability of the counter that is selected is not green. Display your answer on a probability scale.

5. A six-sided spinner shows only the numbers $ 1, 2, 3 $ and $ 4. $ There is an equal chance of it landing on a $ 2 $ or a $ 4, $ but more of a chance of it landing on an odd number than an even number. Select the spinner that satisfies those probability statements.

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Math resources Statistics and probability

Probability distribution

Theo. Prob.

Theoretical probability

Here you will learn about theoretical probability, including probability scales, mutually exclusive events and the probability of something not happening.

Students will first learn about theoretical probability as part of statistics and probability in $7$ th grade and continue to learn about theoretical probability in high school.

What is theoretical probability?

Theoretical probability uses mathematics rather than an experiment to determine the chance of something happening.

To do this, we need to think about two things – the number of times an event may occur and the total number of outcomes available.

$\text{Theoretical probability } =\cfrac{\text{frequency of the event occurring}}{\text{total frequency of possible outcomes}}$

Some events are known as mutually exclusive. This means they cannot happen at the same time. For example, a fair dice cannot land on a $3$ and a $4$ at the same time and so the probability of this event would be $0$ (impossible).

You can, however, find the probability of rolling a $3$ or a $4.$ As these events are mutually exclusive we can add the probabilities together. Here, it is stated that the dice is fair and so the probability of landing on each number is the same.

To calculate the theoretical probability, create a probability model with all the possible outcomes.

The probability of the event occurring can be written as a fraction, decimal or percentage. Use the model to write the fraction.

This is written as $P(3)=\cfrac{1}{6}$ which reads “The probability of rolling a $3$ is $\cfrac{1}{6}.$ ”

Since $P(3)=\cfrac{1}{6}$ and $P(4)=\cfrac{1}{6},$ this means that $P(3$ or $4)=\cfrac{1}{6}+\cfrac{1}{6}=\cfrac{2}{6}=\cfrac{1}{3}.$

Consider this value on a probability scale.

The probability scale goes from $0$ (impossible) to $1$ (certain). This means the probability of something NOT happening must be $1$ subtract the probability of it happening.

$P($ not $A)=1-P(A).$

For example, find the probability of NOT rolling a $3$ or a $4.$

$P($ not $3$ or $4)=1-\left(\cfrac{1}{6}+\cfrac{1}{6}\right)=\cfrac{4}{6}=\cfrac{2}{3}.$

In probability theory, the difference between theoretical and experimental probability is that theoretical probability is based solely on math, whereas experimental probability is based on the outcomes of an experiment and so there is no known probability until the experiment has taken place.

What is theoretical probability?

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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 calculate the theoretical probability

In order to calculate the theoretical probability of an event:

Determine all possible outcomes.
Determine the frequency of the event occurring.
Substitute these values into the formula. Write your answer as a fraction, decimal or percentage.

Theoretical probability 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 vowel? Display it on the probability scale.

Determine all possible outcomes.

There are $11$ cards altogether in the word MATHEMATICS.

2Determine the frequency of the event occurring.

The set of vowels are: $A, E, I, O,$ and $U.$ There are $4$ vowel 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 $4$ and the total frequency of possible outcomes is $11,$ the theoretical probability is

$P($ Vowel $)=\cfrac{4}{11}$

Example 2: rolling a number on a fair die

A fair six-sided die is rolled. What is the probability of it landing on a prime number? Display it on the probability scale.

Determine all possible outcomes.

As the die is six-sided, the total number of outcomes from a single roll is $6.$

Determine the frequency of the event occurring.

The prime numbers on a fair six-sided die are $2, 3,$ and $5.$ Therefore the frequency of prime numbers is $3.$

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

$P($ Prime number $)=\cfrac{3}{6}=\cfrac{1}{2} \ (=0.5)$

Example 3: event not occurring

A bag contains $6$ yellow counters, $3$ blue counters and $1$ white counter. A counter is selected from the bag at random. Find the probability that the counter is not yellow.

Determine all possible outcomes.

The total number of counters in the bag is $6+3+1=10.$

Determine the frequency of the event occurring.

The frequency of yellow counters is $6.$

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

$P($ Yellow $)=\cfrac{6}{10}=0.6$

Since the events are mutually exclusive, the probabilities will all add to $1.$ The probability of not picking a yellow counter is found by subtracting this answer from $1.$

Using decimals,

$P($ Not Yellow $)=1-0.6=0.4.$

Using fractions,

$P($ Not Yellow $)=1-\cfrac{6}{10}=\cfrac{4}{10}.$

Example 4: union of two mutually exclusive events

A bag contains $6$ yellow counters, $3$ blue counters and $1$ white counter. A counter is selected from the bag at random. Find the probability of the counter being yellow or white.

Determine all possible outcomes.

The total number of counters in the bag is $10.$

Determine the frequency of the event occurring.

The frequency of yellow counters is $6$ and the frequency of white counters is $1.$ This means that the frequency of yellow or white counters is $6+1=7.$

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

$P($ Yellow or White $)=\cfrac{7}{10}=0.7$

Example 5: rolling two dice – compound event

Ariel is playing a game and rolls two dice. What is the probability that Ariel rolls two fours?

Determine all possible outcomes.

Use a list to show all possible outcomes.

$\begin{aligned} 1−1 \quad \;\; 2−1 \quad \;\; 3−1 \quad \;\; 4−1 \quad \;\; 5−1 \quad \;\; 6−1 \\ 1−2 \quad \;\; 2−2 \quad \;\; 3−2 \quad \;\; 4−2 \quad \;\; 5−2 \quad \;\; 6−2 \\ 1−3 \quad \;\; 2−3 \quad \;\; 3−3 \quad \;\; 4−3 \quad \;\; 5−3 \quad \;\; 6−3 \\ 1−4 \quad \;\; 2−4 \quad \;\; 3−4 \quad \;\; 4−4 \quad \;\; 5−4 \quad \;\; 6−4 \\ 1−5 \quad \;\; 2−5 \quad \;\; 3−5 \quad \;\; 4−5 \quad \;\; 5−5 \quad \;\; 6−5 \\ 1−6 \quad \;\; 2−6 \quad \;\; 3−6 \quad \;\; 4−6 \quad \;\; 5−6 \quad \;\; 6−6 \end{aligned}$

There are $36$ possible outcomes.

Determine the frequency of the event occurring.

$4\text{−}4$ only appears once in the list of all possible outcomes. The frequency of two fours is therefore $1.$

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

$P($ two fours $)=\cfrac{1}{36} \, (=0.02\bar{7})$

Example 6: flipping a coin three times – compound event

A coin is flipped three times. What is the probability that the coin lands on tails twice and heads once?

Determine all possible outcomes.

Use a tree diagram to show all possible outcomes.

There are $8$ possible outcomes: $TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.$

Determine the frequency of the event occurring.

In the possible outcomes there are Tails-Tails-Heads, Tails-Heads-Tails and Heads-Tails-Tails. The frequency of two tails and one heads is $3.$

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

$P(2$ tails, $1$ head $)=\cfrac{3}{8}.$

Teaching tips for theoretical probability

Introduce theoretical probability with examples that are easy to recreate in the classroom. For example, fill a bag with red marbles and blue marbles and calculate the theoretical probability of picking each color. Or calculate the theoretical probability of a certain number appearing when a dice rolls.

Once students have had enough experience modeling theoretical probability, introduce real life or real world examples of probabilities that involve larger numbers. Give students time to modify their previous strategies to work for larger sample spaces.

Always follow your state standards, but in general it is recommended to introduce independent events first and then dependent events.

Easy mistakes to make

Confusing a number in the sample space for the number of favorable outcomes or total outcomes
For example, a common error when asking for the probability of getting a $2$ on a fair die is $\cfrac{2}{6}.$ It is important to remember that if the die is fair, all outcomes will be equally likely so in this case the probability of getting any of the numbers $1$ to $6$ on a fair die is $\cfrac{1}{6}.$

Subtracting the desired outcome from the possible outcome
Students may incorrectly write the fraction as the frequency of the event occurring out of the remaining frequency.

For example, the probability of rolling a $4$ on a fair six-sided dice is $1$ out of $5$ as there are $5$ numbers remaining. This is incorrect.

The correct answer is $1$ out of $6$ because the probability is equal to the frequency of the event occurring, $1,$ divided by the total number of possible outcomes, $6.$

Practice theoretical probability questions

\cfrac{5}{8}

\cfrac{3}{8}

\cfrac{1}{3}

\cfrac{1}{2}

There are $3$ vowels and $8$ total lettered cards in total and so $P($ Vowel $)=\cfrac{3}{8}.$

\cfrac{3}{7}

\cfrac{2}{10}

\cfrac{3}{10}

\cfrac{1}{3}

There are $3$ square numbers $(1, 4, 9)$ and $10$ numbered cards in total and so $P($ square number $)=\cfrac{3}{10}.$

\cfrac{7}{12}

\cfrac{5}{12}

\cfrac{1}{12}

\cfrac{2}{3}

There are $7$ green or red counters and $12$ counters in total and so

$P($ green or red $)=\cfrac{7}{12}.$

Theoretical Probability 11 US

Theoretical Probability 12 US

Theoretical Probability 13 US

Theoretical Probability 14 US

There are $3$ green counters and $12$ counters in total.

1-\cfrac{3}{12}=1-\cfrac{1}{4}=\cfrac{3}{4}

Theoretical Probability 15 US

Theoretical Probability 16 US

Theoretical Probability 17 US

Theoretical Probability 18 US

There must be the same number of $2$ s and $4$ s. There must be more odd numbers than even numbers and so the remaining values must all contain either a $1$ or a $3.$

\cfrac{1}{15}

\cfrac{2}{3}

\cfrac{1}{4}

\cfrac{6}{15}

Let’s look at all the possible outcomes in a list.

\begin{aligned}&R1−Q1 \quad \;\; R1−Q2 \quad \;\; R1−P \\ &R2−Q1 \quad \;\; R2−Q2 \quad \;\; R2−P \\ &R3−Q1 \quad \;\; R3−Q2 \quad \;\; R3−P \\ &B1−Q1 \quad \;\; B1−Q2 \quad \;\; B1−P \\ &B2−Q1 \quad \;\; B2−Q2 \quad \;\; B2−P \end{aligned}

Notice, the list indicates that there are $3$ different reds, $2$ different blues and $2$ different quarters.

This makes it clear that none of the counters or quarters are being used more than once; just showing that there are multiple.

For example, $R1$ is a different counter from $R2.$ The penny $(P)$ does not need this label, because there is only one.

There are $6$ possible outcomes of a red counter $(R)$ and a quarter $(Q).$

There are $15$ total possible outcomes, so the probability is $\cfrac{6}{15}.$

Theoretical probability FAQs

What are the different types of probability?

There are four: theoretical probability, empirical probability, axiom probability and subjective probability.

What is the experimental probability of an event?

It is the probability result from a preformed experiment. It can be compared to the theoretical probability.

What is conditional probability?

The probability that something will occur, given a previous event, or condition, has already occurred.

See also: Conditional probability

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