Probability means how likely is an event going to occur. In maths, we can express the probability of something happening using either fractions or decimals.

A probability of 1 means that there is 100% chance of that event happening.

E.g. A bag contains only blue balls. Probability of picking a blue ball from that bag must equal to 1 since there is nothing in the bag besides blue balls.

Similarly, a probability of 0.5 or ½ means that the chance of something happening is 50%.

E.g. A bag contains 2 blue balls and 2 white balls. Probability of picking a blue ball from that bag must equal to 0.5.

No. of blue balls = 2

Total number of balls in the bag = 2 + 2 = 4

Probability = (2 ÷ 4) = 0.5 or ½

**Example: **

Julie wins a competition if she picks a red marble from a bag. The bag contains a total of 20 marbles. All marbles in the bag are either red or blue. The bag contains 12 blue marbles.

What is the probability of picking a red marble?

Solution:

We first need to find the number of red marbles in the bag. The bag contains only red marbles and blue marbles.

We know the number of blue marbles = 12

Total number of marbles in the bag = 20

So number of red marbles in the bag = 20 – 12 = 8

Probability of picking a red marble = ⁸⁄₂₀ = ²⁄₅

Let us add a twist to the above question.

Julie manages to pick a red marble from the bag. The marble is not replaced.

She is told that if she picks another red marble, she can win double the prize.

What is the probability of picking a second red marble?

Solution:

The sentence “the marble is not replaced” is key here. “Not replaced” means that the marble that is picked already is neither placed back into the bag nor is it replaced with any other marble.

Therefore, total number of marbles in the bag now = 20 – 1 = 19

Since one marble is already picked, the number of marbles in the bag decreases by one.

Since a red marble was picked, the number of red marbles decreases by 1 too.

So number of red marbles = 8 – 1 = 7

Probability of picking a second red marble = ⁷⁄₁₉

### Tree Diagrams

Tree diagrams are very helpful whilst solving probability questions.

Let us tweak the example above.

Julie wins a competition if she picks a red marble and a blue marble from a bag. The bag contains a total of 20 marbles. All marbles in the bag are either red or blue. The bag contains 12 blue marbles. Once a marble is picked, it is not replaced.

What is the probability of Julie winning?

Solution:

We know number of red marbles = 20 – 12 = 8

Order of picking one red marble and one blue marble could be:

– Red marble first and then a blue marble

– Blue marble first and then a red marble

Let us solve this by using a tree diagram.

Probability of picking red marble first and then blue marble = ⁹⁶⁄₃₈₀

Probability of picking blue marble first and then red marble = ⁹⁶⁄₃₈₀

Julie could win if she does either of these possibilities.

Total probability of winning = ⁹⁶⁄₃₈₀ + ⁹⁶⁄₃₈₀ = ¹⁹²⁄₃₈₀

Tree diagrams are easy to draw during exams if you are stuck. However, do not waste too much time as time is very limited during the exam.