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What Is Probability? (Mathematics Lesson)

What Is Probability?

Probability tells us how likely something is to happen.

Real Examples of Probability

Tossing a Coin

You may have tossed a coin to help you make a decision between two choices.

One person chooses 'Heads', the other 'Tails'. Whoever chooses the side of the coin that faces upwards wins the toss.



The two possible outcomes, or events, are 'Heads' and 'Tails' and we can give the probability of each outcome occuring:

The probability of getting a Heads is:



The probability of getting a Tails is:

Rolling a Die

You may have played games involving rolling dice. An unbiased die has six sides, each labelled with a number of dots, representing the numbers 1 to 6. So there are six outcomes: 1, 2, 3, 4, 5, 6.



The number on the face facing upwards is the result of rolling the die. So, if after rolling the die, one dot faces upwards, this is rolling a 1.

The probability of getting each number can be found:

How to Find the Probability

In general:



Let's apply this formula to the examples of getting a Heads and rolling a 6.

How to Find the Probability of Getting a Heads

Question: What is the probability of getting a Heads when a coin is tossed?

  • Find the 'number of ways it can happen'.

    There is only 1 way of getting a Heads - by the coin landing with the Heads side upwards.

  • Find the 'total number of outcomes'.

    There are 2 possible outcomes - Heads or Tails.

So,

How to Find the Probability of Rolling a 6

Question: What is the probability of rolling a 6 with a fair die?

  • Find the 'number of ways it can happen'.

    There is only 1 way of getting a 6 - the face with 6 dots must face upwards.

  • Find the 'total number of outcomes'.

    There are 6 possible outcomes - 1, 2, 3, 4, 5, or 6.

So,

Other Uses of Probability

Some other examples of where the probability can be found are:

  • the probability of a baby being a boy or a girl.

  • the probability of picking a certain card from a pack.

  • the probability of getting a multiple choice question right or wrong.

A Real Example of How to Find a Probability

The slider below gives a real example of how to find a probability.
Interactive Test
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Note

PROBABILITY

Probability is given as a number between 0 and 1.

A probability of 0 means an event is impossible. The probability of pigs flying is 0 (unless they are put in an aeroplane!).



A probability of 1 means an event is certain. The probability that an elephant is heavier than a fly is 1.

A probability of ½, such as getting a Heads in a coin toss, means it is equally likely to go either way. A coin is as likely to come up Heads as it is to come up Tails.

The closer a probability is to 0 the less likely it is. The closer a probability is to 1 the more likely it is.



WRITING PROBABILITIES

Probability is given as a number between 0 and 1.

This means they are often written as a fraction, such as ½ or ¼.

They may also be written as the equivalent decimal, 0.5 instead of ½ or 0.25 instead of ¼.

A NOTE ON NOTATION

It is often convenient to use a letter to represent an event.

For example, in a coin toss, let:

  • H be the event getting a Heads, and

  • T be the event getting a Tails
Use P to indicate Probability.

Put brackets after the P, and write the letter for the event inside them.

P(H) means the probability of getting Heads.

P(T) means the probability of getting Tails.

HOW OFTEN CAN WE EXPECT SOMETHING TO HAPPEN

Probabilities can be used to find out how often we expect something to happen. This is done by multiplying the number of trials (i.e. how many times you toss a coin, or roll a die) by the probability of an event (i.e. the probability of getting a Heads or a 3).

For example, if you toss a coin 10 times, how many times do you expect to get a Heads?

Multiply the number of tosses, 10, by the probability of getting a Heads, ½.



You would expect to get Heads 5 out of 10 times.

PROBABILITIES ARE CHANCES NOT CERTAINTIES

The probability of getting a Heads in a coin toss is ½. If you tossed a coin 2 times, you would therefore expect to get Heads half the time - 1 Heads and 1 Tails.

Try tossing a coin toss twice. It is very unlikely that you will get this result predicted by probability.

Probability just gives a guide. The more coin tosses you make, the more likely that probability gives the right answer.

If you toss a coin 100 times, you would expect to get Heads half the time, or 50 times. You may get 45 or 55, but it will be near.