GCSE Maths Practice: conditional-probability

Question 4 of 13

This question explores conditional probability when events are independent due to replacement.

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\( \begin{array}{l}\text{A bag contains 10 balls: 6 red and 4 blue.} \\ \text{One ball is drawn at random and then replaced.} \\ \text{What is the probability of drawing a red ball on the second draw, given that the first draw was red?}\end{array} \)

Choose one option:

If an item is replaced, the probability on the next attempt stays the same.

Independent Events and Conditional Probability

This question highlights an important idea in probability: sometimes a condition does not change the probability of an event. This happens when events are independent. Independent events are events where the outcome of one does not affect the outcome of another.

In GCSE Maths, independence is often introduced through situations involving replacement. When an item is replaced after being selected, the sample space remains unchanged. This means the probabilities before and after the first event stay exactly the same.

Understanding the Key Idea

Even though the question uses the phrase "given that", which usually signals conditional probability, replacement makes the condition irrelevant. The bag looks exactly the same before and after the first draw. As a result, the probability on the second draw is identical to the probability on the first draw.

Step-by-Step Method

  1. Check whether the item is replaced or not.
  2. If the item is replaced, the total number of outcomes stays the same.
  3. Count the favourable outcomes.
  4. Write the probability using the unchanged total.

Worked Example 1

A box contains 8 pens: 5 black and 3 blue. One pen is chosen and then replaced. What is the probability the next pen chosen is black?

Answer: The box still contains 8 pens, 5 of which are black. The probability is \(\frac{5}{8}\).

Worked Example 2

A spinner has 4 equal sections: 1 red and 3 green. It is spun twice. What is the probability of landing on green on the second spin, given that the first spin was green?

Answer: Each spin is independent. The probability of green remains \(\frac{3}{4}\).

Common Mistakes to Avoid

  • Changing the total when the item is replaced.
  • Assuming all conditional probabilities must change.
  • Confusing independent and dependent events.
  • Reducing the number of favourable outcomes incorrectly.

Real-Life Application

Independent events appear frequently in real life. Examples include tossing a coin multiple times, rolling a die, or selecting an item from stock that is immediately restocked. In all of these cases, previous outcomes do not influence future probabilities.

Frequently Asked Questions

Is this still conditional probability?
Yes. The question includes a condition, but it demonstrates that the condition does not always affect the probability.

How can I quickly tell if events are independent?
Look for replacement, restocking, or repeated actions like spins or rolls.

Do I need a formula here?
No. At Foundation level, understanding the situation is more important than using formal notation.

Study Tip

If the question says the item is replaced, pause and remind yourself: nothing has changed. This often makes the answer much simpler than it first appears.

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