## Probability I – The Basics

Probability is the study of chance. What are the chances are a certain event occurring? How likely is it to happen again? These are all questions involved in probability.

Probability is widely used across many areas today such as finance, statistics, gambling and even physics!

$\mathrm{Probability} = \frac{\mathrm{Number \; of \; possible \; successes}}{\mathrm{Total \; number \; of \; possible \; outcomes}}$

For an specified event A, we often write “The probability of A” as $P(A)$.

For example, the probability of rolling a dice and getting an even number would be the number of possible successes, 2, 4 and 6, divided by the total number of possible outcomes, 1, 2, 3, 4, 5 and 6. So $P(\mathrm{Even\; number}) = \frac{3}{6} = \frac{1}{2}$.

If something is certain to happen, we say this has a probability of 1.

If something is impossible, then we say this has a probability of 0.

It follows then that the probability of something not happening is 1 minus the probability that it will happen.

Did you know? – The exact history of probability is uncertain, but we know that the study of chance has been linked to gambling for thousands of years!

Example 1

There are 20 balls in a bag; 7 are red, 5 are yellow, 6 are green and 2 are blue. What is the probability of not picking a green ball?

Using the formula above, the number of green balls is 6, and total number of balls is 20, so $P(\mathrm{Not \; green}) = 1 - \frac{6}{20} = 0.7$.

Example 2

There’s another bag full of coloured balls, red, yellow, green and blue. It is not known how many of each colour are in the bag. The balls are picked out at random and replaced. You do this 1000 times and get the following results:

• 500 red balls picked
• 200 yellow balls picked
• 150 green balls picked
• 150 blue balls picked

i) What is the probability of picking a yellow ball?
ii) If there are 200 balls in the bag, how many are likely to be blue?

i) For the 1000 balls picked out, 200 were yellow. This gives us $P(\mathrm{Yellow}) = \frac{200}{1000} = \frac{1}{5} = 0.2$.

ii) Although we don’t know the exact number of balls in the bag, we can predict the numbers from the ones we have already picked out. We know that $P(\mathrm{Blue}) = \frac{150}{1000} = 0.15$. So if there’s 200 balls in the bag, the expected number of blue balls is $0.15 \times 200 = 30$.

Now take a look at our article on Probability – Multiple Events.