Probability

Mathematically, probability of an outcome can be expressed as follows:

\(Probability = \frac{successful \ outcomes}{Total \ outcomes}\)

Note:
First count the number of outcomes you are looking for and divide it by the number of outcomes in the event.

Now, let's take a look at some examples to help us understand the formula above:

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Example 1

A fair die is thrown once. What is the probability that the number obtained is a factor of 6?

Solution

When a die is thrown the possible outcomes are \(\{1, 2, 3, 4, 5, 6\}\). This means that there are 6 total possible outcomes.

Can you list the factors of 6?

They are {1, 2, 3 and 6}, which consists of 4 outcomes.

Factors of a number are the numbers that can divide the given number without getting a remaider.

It implies that,

Prob (factor of 6) \(= \frac{4 \ outcomes}{6 \ possible \ outcomes}\)

Prob (factor of 6) \(= \frac{4}{6}\)

Prob (factor of 6) \(=\frac{2}{3}\)

\(\therefore\) the probability of obtaining a factor of 6 is \(\frac{2}{3}\)

Note:
Remember, that it is a good practice to express fractions in their simplest term always.

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Example 2

What is the probability of obtaining a factor of 8 when a die is thrown?

Solution

Possible outcomes \(\Rightarrow\) {1, 2, 3, 4, 5, 6}

Factors of 8 within the possible outcomes \(\Rightarrow\) {1, 2, 4}

P(factor of 8) \(= \frac{3}{6} \)

P(factors of 8) \(= \frac{1}{2}\)

\(\therefore\) The probability of obtaining a factor of 8 is \(\frac{1}{2}\)

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Example 3

The probability of obtaining a head when a coin is tossed is \(\frac{1}{2}\). Convert this to a percentage.

Solution

Prob (Head) \(= \frac{1}{2} \)

Converting to a percentage:

\(\Rightarrow \frac{1}{2} \times 100\%\)

\(\Rightarrow \frac{100}{2}\%\)

\(\Rightarrow 50\%\)

\(\therefore\) the probability of obtaining a head when a coin is tossed is \(50\%\)

Remember that to convert from a fraction to decimal, you need to divide the numerator by the denominator. You can achieve this by applying long division.

Also, to convert from percentage to a decimal, you need to divide by 100.

Fractions can be expressed as ratios as well.