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AQA GCSE · Question 13 · Probability
A bag contains one £5 note, one £10 note, one £20 note and one £50 note. Amaan picks two of the notes at random without replacement. Work out the probability that he has picked at least £30.
A bag contains one £5 note, one £10 note, one £20 note and one £50 note. Amaan picks two of the notes at random without replacement. Work out the probability that he has picked at least £30.
How to approach this question
1. Determine the total number of possible outcomes. Since two notes are picked without replacement from four, think about how many choices there are for the first pick and then the second.
2. Identify what "at least £30" means (the sum is £30 or more).
3. List all the possible pairs of notes that can be picked.
4. For each pair, calculate the sum.
5. Count how many of these pairs have a sum of £30 or more. This is the number of favourable outcomes.
6. Calculate the probability by dividing the number of favourable outcomes by the total number of outcomes.
Full Answer
There are 4 notes in total. The number of ways to pick 2 notes without replacement is 4 × 3 = 12 possible ordered pairs.
We want the probability that the sum of the two notes is at least £30. It's easier to list the pairs that satisfy this condition.
The possible pairs are (Note 1, Note 2).
- If Note 1 is £5, Note 2 can be £50. Sum = £55. (Favourable)
- If Note 1 is £10, Note 2 can be £20 or £50. Sums = £30, £60. (Favourable)
- If Note 1 is £20, Note 2 can be £10, £50. Sums = £30, £70. (Favourable)
- If Note 1 is £50, Note 2 can be £5, £10, £20. Sums = £55, £60, £70. (Favourable)
Let's list all favourable ordered pairs:
(£5, £50), (£10, £20), (£10, £50), (£20, £10), (£20, £50), (£50, £5), (£50, £10), (£50, £20).
There are 8 favourable outcomes.
The total number of possible outcomes is 4 choices for the first pick and 3 for the second, so 4 × 3 = 12.
Probability = (Favourable Outcomes) / (Total Outcomes) = 8 / 12 = 2/3.
This is a probability problem without replacement.
First, let's find the total number of ways Amaan can pick two notes.
There are 4 choices for the first note.
After picking one, there are 3 choices left for the second note.
Total number of ordered pairs (outcomes) = 4 × 3 = 12.
Next, let's find the number of pairs where the sum is at least £30 (i.e., £30 or more). We can list them systematically:
- (£5, £10) = £15 (No)
- (£5, £20) = £25 (No)
- (£5, £50) = £55 (Yes)
- (£10, £5) = £15 (No)
- (£10, £20) = £30 (Yes)
- (£10, £50) = £60 (Yes)
- (£20, £5) = £25 (No)
- (£20, £10) = £30 (Yes)
- (£20, £50) = £70 (Yes)
- (£50, £5) = £55 (Yes)
- (£50, £10) = £60 (Yes)
- (£50, £20) = £70 (Yes)
Counting the "Yes" outcomes, we have 8 favourable outcomes.
The probability is the number of favourable outcomes divided by the total number of outcomes.
P(sum ≥ £30) = 8 / 12
This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 4.
P(sum ≥ £30) = 2 / 3.
Common mistakes
✗ Forgetting that the order matters when listing outcomes, e.g., considering (£10, £20) but not (£20, £10). This leads to an incorrect total number of outcomes.
✗ Calculating with replacement (total outcomes would be 4x4=16).
✗ Misinterpreting "at least £30" as "more than £30".
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