ExpertMedium3 marksStructured
AQA GCSE · Question 21 · Probability
A bag contains 25 discs. 11 are red, 9 are blue and 5 are yellow. Ashley picks three of the discs at random without replacement. Ashley's first disc is red. Work out the probability that all three discs are different colours.
A bag contains 25 discs. 11 are red, 9 are blue and 5 are yellow. Ashley picks three of the discs at random without replacement. Ashley's first disc is red. Work out the probability that all three discs are different colours.
How to approach this question
1. The first pick is already known to be red. This is a conditional probability problem.
2. After the first red disc is picked, how many discs of each colour are left? How many total discs are left?
3. For all three discs to be different colours, the second and third picks must be blue and yellow.
4. There are two possible orders for this: (Red, Blue, Yellow) and (Red, Yellow, Blue).
5. Calculate the probability of the second pick being blue AND the third being yellow. Remember the number of discs changes after each pick.
6. Calculate the probability of the second pick being yellow AND the third being blue.
7. Add the probabilities from steps 5 and 6 together for the final answer.
Full Answer
Given that the first disc is red, there are now 24 discs left in the bag.
The remaining discs are: 10 red, 9 blue, 5 yellow.
We want the next two discs to be different colours from red and from each other. This means the next two picks must be (Blue, Yellow) or (Yellow, Blue).
Case 1: The second disc is Blue, and the third is Yellow.
- Probability of picking a blue disc second: There are 9 blue discs out of 24 total. P(2nd is Blue) = 9/24.
- After picking a blue disc, there are 23 discs left, including 5 yellow ones.
- Probability of picking a yellow disc third: P(3rd is Yellow | 2nd is Blue) = 5/23.
- Probability of this sequence (R, B, Y) = (9/24) * (5/23) = 45/552.
Case 2: The second disc is Yellow, and the third is Blue.
- Probability of picking a yellow disc second: There are 5 yellow discs out of 24 total. P(2nd is Yellow) = 5/24.
- After picking a yellow disc, there are 23 discs left, including 9 blue ones.
- Probability of picking a blue disc third: P(3rd is Blue | 2nd is Yellow) = 9/23.
- Probability of this sequence (R, Y, B) = (5/24) * (9/23) = 45/552.
The total probability is the sum of the probabilities of these two mutually exclusive cases:
Total P = P(R, B, Y) + P(R, Y, B) = 45/552 + 45/552 = 90/552.
Simplifying the fraction:
90/552 = 45/276 = 15/92.
This is a conditional probability problem because we are given information about the first event: "Ashley's first disc is red."
Step 1: Update the contents of the bag.
Originally: 11R, 9B, 5Y (Total 25)
After one red is picked: 10R, 9B, 5Y (Total 24)
Step 2: Determine the required outcomes for the next two picks.
For all three discs to be different colours, and knowing the first is red, the next two must be one blue and one yellow. There are two possible orders for this to happen:
- Order 1: Second is Blue, Third is Yellow (B, Y)
- Order 2: Second is Yellow, Third is Blue (Y, B)
Step 3: Calculate the probability of each order.
- P(Order 1: B, Y) = P(2nd is Blue) × P(3rd is Yellow, given 2nd was Blue)
= (9/24) × (5/23) (After picking a blue, 5 yellow are left out of 23)
= 45/552
- P(Order 2: Y, B) = P(2nd is Yellow) × P(3rd is Blue, given 2nd was Yellow)
= (5/24) × (9/23) (After picking a yellow, 9 blue are left out of 23)
= 45/552
Step 4: Add the probabilities of the two possible orders.
Since either order satisfies the condition, we add their probabilities:
Total Probability = P(B, Y) + P(Y, B) = 45/552 + 45/552 = 90/552.
Step 5: Simplify the fraction.
Both 90 and 552 are divisible by 6.
90 ÷ 6 = 15
552 ÷ 6 = 92
So the probability is 15/92.
Common mistakes
✗ Forgetting that the first pick is a given condition and calculating the probability from the start, e.g., (11/25) * (9/24) * (5/23).
✗ Only calculating the probability for one order (e.g., R, B, Y) and not considering the other (R, Y, B).
✗ Incorrectly updating the number of discs after each pick (e.g., keeping the denominator as 24 for the third pick).
Practice the full AQA GCSE Maths Higher Tier Paper 3 Calculator
32 questions · hints · full answers · grading
More questions from this exam
Q01Work out the reciprocal of 10/3. Give your answer as a decimal.EasyQ02.1The table shows information about the number of houses with solar panels in a town. Complete the ...EasyQ02.2Use the graph to estimate the number of houses with solar panels in 2023.EasyQ03.1A building in the shape of a cylinder has diameter 40 m and height 55 m. On the centimetre grid, ...MediumQ03.2On this centimetre grid, draw the front elevation of the building. Use a scale of 1 cm to 10 m.Medium