Beginning Probability: A Gentle Introduction to the Mathematics of Chance

Beginning Probability for Statistics

In this lesson, you’ll see what beginning probability means, then you’ll learn how to work out probability in various cases.

Likely and Unlikely

Some events are more likely than others to happen. For example, it may be more likely to rain tomorrow than snow tomorrow.

Mathematics can express how likely things can happen by having a scale that goes from 0 to 1, where 0 means the event will not happen and 1 means the event must happen.


An example of an impossible event is finding a square with unequal sides.

An example of a certain event is finding a triangle with three sides.

These are mathematical examples; in everyday life, events fall between these two extremes.

Tossing a Coin

If I toss a coin how likely is it to land with heads facing up?

Assuming it’s a fair coin, which means that heads and tails are equally likely to come up, the answer will be somewhere between impossible and certain.

Or, somewhere between 0 and 1 on the scale above.

In fact since the coin is fair, the chance of heads will be ½.  (And the chance of tails will also be ½).


The probability of an event occurring is defined as a fraction where the numerator is the number of ways in which an event can happen, and the denominator is the total number of all possible outcomes.



So in the case of throwing a coin there are 2 possible outcomes (heads and tails). So 2 is the denominator of the probability fraction.

And as you want the probability of heads, and there is only 1 way you can get heads, the numerator of the probability fraction is 1.

So the probability of getting heads is 1/2 .

Throwing a Die

Here’s another example. What is the probability of getting five when throwing a die?

Well, there are 6 possible outcomes and the five comes up in only 1 of these. So the answer is1/6 .

You could say there’s 1 chance in 6 of getting a five.

Choosing a Vowel

What is the probability of choosing a vowel from a bag containing all the letters of the alphabet (only one of each)?

There are 26 possible outcomes and 5 of these are vowels so the probability is 5/26 .

Three Chances

And with that same bag of letters, what is the probability of choosing a P or a Q or an R?

This time there are 3 chances of success out of 26 so the answer is 3/26 .


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