GCSE Maths Practice: listing-outcomes

Understanding Basic Probability with Coloured Balls

This type of question is a classic introduction to probability at GCSE Foundation level. Probability measures how likely something is to happen and is written as a fraction, decimal, or percentage. In simple situations like choosing a ball from a small bag, probability is calculated using one key idea: compare the number of favourable outcomes to the total number of possible outcomes.

Here, the favourable outcomes are the red balls because that is the event we are interested in. The total outcomes are all the balls in the bag, regardless of colour. When you divide the favourable outcomes by the total, you get a probability between 0 and 1. A result close to 1 means the event is very likely, while a result close to 0 means the event is unlikely.

Step-by-Step Method

1. Count the total number of balls in the bag. In this case, 3 red + 2 blue = 5 total.
2. Identify how many balls match the event described, which is picking a red ball. There are 3 red balls.
3. Write the probability as a fraction: favourable outcomes ÷ total outcomes.
4. Simplify the fraction if possible. In this case, the fraction is already simplified.

Using these steps makes it easier to apply probability rules consistently, even in more difficult questions later on.

Worked Example 1: Picking a Blue Ball

If the same bag is used and the question asks for the probability of picking a blue ball, the favourable outcomes would be 2 instead of 3. The probability becomes 2/5. This example shows how changing the event changes the favourable count but the method remains identical.

Worked Example 2: Picking a Red Ball from a Larger Bag

Imagine a bigger bag containing 4 red balls, 3 blue balls, and 3 yellow balls. The total is 10 balls. If the event is picking a red ball, you divide: 4 favourable outcomes by 10 total outcomes to get 4/10, which simplifies to 2/5.

Worked Example 3: Picking an Even Number from Number Cards

A set of number cards contains 1, 2, 3, 4, and 5. The favourable outcomes (even numbers) are 2 and 4, so there are 2 favourable outcomes out of 5 total. This leads to a probability of 2/5. Even though the context has changed, the method remains exactly the same.

Common Mistakes

• Miscounting the total number of items. Students sometimes count only the coloured items relevant to the event, forgetting that all items make up the total.
• Swapping favourable and total numbers. Writing total on top and favourable on the bottom produces a value greater than 1, which is impossible in probability.
• Not simplifying fractions. While not required here, many exam questions expect simplified answers.

Real-Life Applications

Simple probability appears in everyday decision-making. Choosing a sweet from a mixed bag, selecting a random card from a stack, or estimating the chance of rain all use the same mathematical principles. In science, probability is used in genetics to predict traits. In computing, random selection is used in simulations, games, and encryption. Understanding basic probability builds confidence for more applied situations later in school or work.

FAQ

Q: Can probability ever be greater than 1?
A: No. A probability must always be between 0 and 1.

Q: What if all balls in the bag were red?
A: The probability of picking a red ball would be 1, meaning it is certain.

Q: What if no red balls were in the bag?
A: The probability would be 0, meaning it is impossible.

Study Tip

Always begin by writing down the favourable and total outcomes before forming the fraction. This removes confusion and helps prevent common exam errors.