Probability Depth
Probability Depth goes beyond simple single-event chance into combined-event probability: finding the probability of two independent events both happening (such as a spinner and a dice, or two picks with replacement), and using 'one minus' to find the probability an event does NOT happen. It is tested in GL Assessment and CEM-style numerical reasoning sections, where probability is expressed interchangeably as a fraction, decimal or percentage.
Before you start
Make sure you're comfortable with these topics first:
Method
- Identify the total number of equally likely outcomes for a single event before finding any probability.
- Write a probability as favourable outcomes over total outcomes, then simplify the fraction fully.
- For 'the event does NOT happen', use P(not A) = 1 - P(A) rather than counting the unfavourable outcomes separately when a direct count is harder.
- For two independent events happening together, such as two spinners, or a coin and a dice, multiply their individual probabilities.
- For 'at least one' of two events, it is often easier to find the probability that neither happens and subtract from 1, rather than listing every successful combination.
- Convert a probability freely between a fraction, a decimal and a percentage, depending on what the question asks for.
- Remember every probability must be between 0 (impossible) and 1 (certain), and use this to sanity check any answer.
Worked example
A fair coin is flipped and a fair six-sided dice is rolled at the same time. What is the probability of getting heads on the coin AND a number greater than 4 on the dice?
- Find the probability of heads on the coin: P(heads) = 1/2.
- Find the numbers greater than 4 on the dice: 5 and 6, so 2 out of 6 outcomes.
- Find the probability of a number greater than 4: 2/6 = 1/3.
- Since the coin and dice are independent, multiply the two probabilities: 1/2 x 1/3.
- Calculate: 1/2 x 1/3 = 1/6.
- State the answer: the probability is 1/6.
Practice questions
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Q1A bag has 4 red and 6 blue counters. What is the probability of picking a red counter?Show answer
Answer: 4/10 = 2/5.
Q2Using the bag above, what is the probability of NOT picking a red counter?Show answer
Answer: 1 - 2/5 = 3/5.
Q3A fair dice is rolled. What is the probability of rolling an even number?Show answer
Answer: 3/6 = 1/2.
Q4A spinner has 5 equal sections numbered 1 to 5. What is the probability of NOT spinning a 3?Show answer
Answer: 1 - 1/5 = 4/5.
Q5Two fair coins are flipped. What is the probability that both land on tails?Show answer
Answer: 1/2 x 1/2 = 1/4.
Q6A box has 10 tickets numbered 1 to 10. What is the probability of picking a multiple of 3?Show answer
Answer: The multiples of 3 up to 10 are 3, 6 and 9, so the probability is 3/10.
Q7Write the probability 0.2 as a fraction in its simplest form.Show answer
Answer: 2/10 = 1/5.
Q8A probability is given as 3/4. Write this as a percentage.Show answer
Answer: 75%.
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A bag contains 3 red, 4 blue and 5 green counters. A counter is picked at random, its colour noted, then it is put back in the bag before a second counter is picked at random. (a) Find the probability that both counters picked are red. (b) Give your answer as a decimal.
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The probability that it rains tomorrow is 0.3. Find the probability that it does NOT rain tomorrow, giving your answer as a fraction in its simplest form.
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Free printable worksheet
Want more practice on paper? Download the probability depth worksheet pack - 9 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 22 of 11+ Maths Workbook 3, the whole course as one free printable PDF.
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