Math resources Statistics and probability

Probability

How to find probability

# How to find probability

Here you will learn about how to find probability, including how to find probability using words, recognising the likelihood of an event and placing events on a probability scale.

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

## What is how to find probability?

Finding probability is when you make judgements as to whether an event will take place. To do this, you describe how probable the event is.

For example,

You may say that it is likely to rain tomorrow, or that it is impossible for the sun not to rise.

Some other common words used to describe the probability of an event happening include; certain, very likely, even chance, unlikely and very unlikely.

These words can be placed on a probability scale starting at 0 (impossible) and ending at 1 (certain).

### Probability scale

The probability scale is used to represent probabilities on a number line from 0 (impossible) to 1 (certain).

Events that are neither impossible nor certain, can be placed somewhere on a probability scale. To do this, you need to consider the words suitable to describe the probability of the event happening, or the number which would represent the chance of the event happening.

For example,

Step-by-step guide: Probability scale

The probability of an event happening can be calculated using the formula

\text{Probability}=\cfrac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}

For example,

Let’s calculate the probability of getting an odd number when a fair die is rolled.

The desired outcome is rolling an odd number. There are 3 odd numbers on the die.

The total number of possible outcomes is 6 since there are 6 numbers on the die.

The probability of getting an odd number =\cfrac{\text { Number of desired outcomes }}{\text { Total number of outcomes }}=\cfrac{3}{6}

Step-by-step guide: How to calculate probability

## 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.

## How to describe probability in words

In order to describe probability in words:

1. Consider the likeliness of the event happening.
2. Choose an appropriate word from the list to describe the probability.

## How to find probability in words examples

### Example 1: how to find probability in words

Describe the likelihood that a number 7 will be scored when an ordinary six-sided dice is rolled once.

1. Consider the likeliness of the event happening.

An ordinary six-sided dice will have the numbers 1 to 6 on its faces. Scoring a 7 on an ordinary six-sided dice will definitely not happen.

2Choose an appropriate word from the list to describe the probability.

Scoring a 7 on an ordinary six-sided dice cannot happen.

It is impossible to score a 7 on an ordinary six-sided dice.

### Example 2: how to find probability in words

In words, describe the probability that a red counter is obtained when a counter is taken at random from a bag that contains 11 red counters and 2 blue counters.

Consider the likeliness of the event happening.

Choose an appropriate word from the list to describe the probability.

## How to place events on a probability scale

In order to place events on a probability scale:

1. Consider the mathematical likeliness of the event happening.
2. Choose the appropriate place on the scale to place the event.

## Placing events on a probability scale examples

### Example 3: placing events on a probability scale

Mark on the scale below, the probability that an ordinary six-sided dice is rolled and lands on an even number.

Consider the mathematical likeliness of the event happening.

Choose the appropriate place on the scale to place the event.

### Example 4: placing events on a probability scale

There are 12 marbles in a bag.

6 marbles are yellow.

4 marbles are green.

2 marbles are red.

The probability that Lucas randomly chooses a red marble is \cfrac{2}{12} or \cfrac{1}{6}.

On the probability scale, mark the probability.

Consider the mathematical likeliness of the event happening.

Choose the appropriate place on the scale to place the event.

## How to find probability

In order to find probability:

1. Identify the sample space.
2. Write out the basic probability as a ratio.

## How to find probability examples

### Example 5: basic probability of a single event

Jamie has the following cards:

A card is chosen at random. Find the probability the card has a letter B on it.

Identify the sample space.

Write out the basic probability as a ratio.

### Example 6: probability example of multiple events

Hussain rolls a fair dice and flips a fair coin. What is the probability that the dice lands on 3 and the coin lands on tails?

Identify the sample space.

Write out the basic probability as a ratio.

### Teaching tips for how to find probability

• When introducing finding probability, start with simple events before moving on to multiple events. Provide many opportunities for students to conduct their own experiments (flipping coins, rolling dice, etc.).

This allows students to create relative frequency tables and graphs and make sense of them. Also, encourage students to represent the sample space in an increasingly efficient way.

• Give students who need extra support access to a probability calculator to check their work.

### Easy mistakes to make

• Forgetting the probability scale goes from \bf{0} (impossible) to \bf{1} (certain)
Remember that impossible is at 0 and certain is at 1, when probability scales are marked numerically rather than with words.

For example,

• Not understanding even chance
An event can only be described as having an “even chance” of occurring if the number of times it could occur is exactly the same as the other possible outcome.

For example,
When flipping a coin, a coin only has two sides. One contains heads and the other tails. So the coin landing on heads and the coin landing on tails has an equal chance of happening.

• Not understanding unfair events
Just because an event has two outcomes, it does not mean that the two events have equal probabilities.

For example,
A cube has six faces. One face is red and the other 5 faces are blue. There are 2 outcomes: red and blue. But the probabilities are different.

The probability of getting red is \cfrac{1}{6} and the probability of getting blue is \cfrac{5}{6}.

### Practice how to find probability questions

1. Place the words in the box onto the probability scale.

Remember that impossible is at 0 and certain is at 1, when probability scales are marked numerically rather than with words.

For example,

2. Use a word from the box to describe the probability that a blue marble is picked from the bag below.

Impossible

Unlikely

Certain

Likely

There are 3 blue marbles and 2 that are not blue. This means the probability is slightly greater than 0.5, so it is a likely event.

3. Use a word from the box to describe the probability that a red marble is picked from the bag below.

Unlikely

Impossible

Very Unlikely

Likely

There are no red counters in the bag, so a red counter cannot be picked. The probability is 0.

4. The probability that a green marble is picked from the bag below is \cfrac{1}{5}. Show where this belongs on the probability scale.

The blue number line below shows fifths.

\cfrac{1}{5} \, (or \cfrac{2}{10})  is close to 0, so it is unlikely.

5. Alyssandra uses some letter cards to spell out her name.

She takes one of these cards at random.

Mark on the scale below, the probability that Alyssandra takes a card containing the letter A.

\text{Probability}=\cfrac{\text{Number of desired outcomes}}{\text{Total number of outcomes}}

The total number of outcomes is 10, because of the 10 different cards. The desired outcome is the letter A, which is on 3 cards.

So the probability as a ratio is 3 to 10 or \cfrac{3}{10}, which is also equal to 0.3 as a decimal. This falls between 0 and 0.5 on the probability scale, making it an unlikely event.

6. Maria flips a fair coin and chooses a marble from the bag. What is the probability that the coin lands on heads and the marble is blue?

\cfrac{3}{10}

\cfrac{1}{6}

\cfrac{1}{2}

\cfrac{3}{5}

Let’s model with a table.

Heads and blue (HB) appear 3 times together in the table, so the number of favorable outcomes is 3. The number of total possible outcomes is 10.

P(\text { Heads and Blue })=\cfrac{3}{10}

## How to find probability FAQs

What is theoretical probability?

The use of math to calculate the probability in theory.

What are dependent events?

An event that is affected by the outcome of the first event.

What does a probability above \bf{0} mean?

It means it has at least one favorable outcome.

How does probability use statistics?

To understand the probability of an unknown structure, you must use statistics. This can include but is not limited to, using the normal distribution, standard deviation, confidence intervals, z -scores, z -tables and other advanced statistical processes and theorems.

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