GCSE Maths Practice: probability-scale

Understanding Probability with Mixed Objects

This problem involves drawing two cards in sequence from a mixed bag containing yellow, blue, and green cards. Questions like this are excellent practice for understanding dependent probability, where the outcome of one event influences the next. Since the cards are drawn without replacement, the total number of cards changes after the first draw, which means the sample space for the second draw is different. Being able to track these changes is an essential skill in GCSE Maths, particularly when working with multi-step probability problems, tree diagrams, or real-world sampling situations.

The bag contains 3 yellow cards, 5 blue cards, and 4 green cards, making a total of 12. To find the probability of drawing a yellow card first, simply note that there are 3 favourable outcomes out of 12 possible outcomes. Once that card has been removed from the bag, there are now only 11 cards left. The number of blue cards remains unchanged at 5, so the probability of drawing a blue card second becomes 5 out of 11.

Step-by-Step Method

1. Identify the total number of items in the bag: 12 cards.
2. Find the probability of drawing a yellow card first: \(3/12\).
3. Remove the drawn card, reducing the total to 11.
4. Count the number of blue cards still in the bag: 5.
5. Find the probability of drawing a blue card second: \(5/11\).
6. Multiply the two probabilities to calculate the combined probability.

Worked Example 1

If you wanted the probability of drawing a blue card first and then a yellow card, the steps would be similar. P(blue first) = \(5/12\), and after that draw, P(yellow second) = \(3/11\). The combined probability becomes \((5/12) \times (3/11)\).

Worked Example 2

Suppose the question asked for drawing two yellow cards in a row without replacement. First, P(yellow) = \(3/12\). After removing one yellow card, there are now 2 yellow cards left out of 11 total cards, so the second probability is \(2/11\). The combined probability is \((3/12) \times (2/11)\).

Common Mistakes

• Forgetting to reduce the total number of cards from 12 to 11 after the first draw.
• Incorrectly reducing the number of blue cards after drawing a yellow card. Only the total changes; blue cards remain the same unless a blue card is drawn.
• Adding probabilities instead of multiplying them. Sequential dependent events always require multiplication.
• Confusing the order of events. The question asks specifically for yellow first and blue second.

Real-Life Applications

This type of probability appears in many practical contexts. For example, if you inspect products from a batch, the quality of the first item affects the remaining pool. In medical sampling, removing one patient from a group changes the probability structure for the next selection. Even everyday decisions like choosing sweets from a mixed bag follow the same logical pattern. Learning how to calculate these probabilities accurately builds the foundation for more advanced statistical reasoning later on.

FAQ

Q: Why does the second probability use 11 instead of 12?
A: Because the first card is removed, reducing the total number of cards.

Q: Why multiply the two probabilities?
A: Because both events must occur in order—this is the rule for sequential 'and' events.

Q: Does the order matter?
A: Yes. Yellow followed by blue is different from blue followed by yellow.

Study Tip

Always write down the new totals after each draw. This prevents the most common mistakes and ensures accuracy in multi-step probability questions.