About Probability
Probability is a count of the favourable outcomes divided by a count of all the equally likely outcomes. The formula is never the difficulty. The difficulty is making sure that the favourable outcomes and the total are counted over the same set of possibilities.
What you need to understand
- Probability = number of favourable outcomes / total number of equally likely outcomes.
- Every probability lies between 0 and 1, so an answer outside that range is wrong on inspection.
- The total is the size of the whole sample space, not the number of favourable outcomes.
- Two dice give 36 equally likely outcomes, n coins give 2^n, and a pack of cards gives 52.
- The probability of an event and the probability of its complement add up to 1.
- At least and exactly are different questions. At least two heads includes the case of three heads, while exactly two does not.
Formulas to remember
How to work through these questions
- Count the total number of possible outcomes first and write it down as the denominator.
- Count the favourable outcomes carefully, listing them when the number is small.
- Reduce the fraction at the end and leave it as a fraction rather than turning it into a decimal.
- For threshold questions such as at least three, it is often quicker to count the complement and subtract it from 1.
- Check that the answer lies between 0 and 1 before choosing an option.
Mistakes that cost marks
- Putting the favourable outcomes over the number of other outcomes instead of over the total.
- Assuming outcomes are equally likely when they are not, such as treating every total from 2 to 12 as equally likely.
- Reading at least as though it said exactly.
- Counting only one arrangement when several arrangements give the same result.
- Leaving the fraction unreduced when the options are given in lowest terms.
Worked example
Two dice are thrown together. What is the probability that the total is 8?
- The total number of outcomes is 6 x 6 = 36.
- The pairs giving a total of 8 are (2,6), (3,5), (4,4), (5,3) and (6,2), which is 5 outcomes.
- Probability = 5/36.
- Check: the answer lies between 0 and 1, and since 7 is the most likely total with 6 outcomes, 5 outcomes for 8 is sensible.
Answer: 5/36
Practice questions with answers
A few Probability questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
From a bag containing 9 red, 6 blue and 3 green balls, a single ball is drawn. What is the probability of drawing a red ball?
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A
$\dfrac{5}{6}$
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B
$\dfrac{1}{6}$
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C
$\dfrac{1}{2}$
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D
$\dfrac{1}{3}$
Answer: Option C — with explanation
Total number of balls = 9 + 6 + 3 = 18.
The number of favourable outcomes is the number of red balls, 9.
Probability = $$\dfrac{1}{2}$$.
Common mistakes
- added two of the colours together: $\dfrac{5}{6}$
- assumed the three colours are equally likely: $\dfrac{1}{3}$
- gave the probability of the third colour: $\dfrac{1}{6}$
Question 2
A bag contains 4 red, 6 blue and 3 green balls. One ball is drawn at random. What is the probability that it is red?
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A
$\dfrac{10}{13}$
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B
$\dfrac{4}{9}$
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C
$\dfrac{4}{13}$
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D
$\dfrac{6}{13}$
Answer: Option C — with explanation
Total number of balls = 4 + 6 + 3 = 13.
The number of favourable outcomes is the number of red balls, 4.
Probability = $$\dfrac{4}{13}$$.
Common mistakes
- added two of the colours together: $\dfrac{10}{13}$
- gave the probability of a different colour: $\dfrac{6}{13}$
- compared the red balls with the other colours instead of with the total: $\dfrac{4}{9}$
Question 3
A bag holds 5 red, 8 blue and 4 green balls. If one ball is taken out without looking, find the probability that it is red.
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A
$\dfrac{5}{17}$
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B
$\dfrac{4}{17}$
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C
$\dfrac{5}{12}$
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D
$\dfrac{8}{17}$
Answer: Option A — with explanation
Total number of balls = 5 + 8 + 4 = 17.
The number of favourable outcomes is the number of red balls, 5.
Probability = $$\dfrac{5}{17}$$.
Common mistakes
- compared the red balls with the other colours instead of with the total: $\dfrac{5}{12}$
- gave the probability of the third colour: $\dfrac{4}{17}$
- gave the probability of a different colour: $\dfrac{8}{17}$
Question 4
4 coins are tossed together. What is the probability of getting at least four heads?
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A
$\dfrac{1}{8}$
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B
$\dfrac{15}{16}$
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C
$\dfrac{1}{2}$
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D
$\dfrac{1}{16}$
Answer: Option D — with explanation
4 coins give 2^4 = 16 equally likely outcomes.
The number of outcomes with at least four heads is 1.
Probability = $$\dfrac{1}{16}$$.
Common mistakes
- counted the outcomes that do not qualify: $\dfrac{15}{16}$
- counted one outcome too many: $\dfrac{1}{8}$
- assumed half of the outcomes qualify: $\dfrac{1}{2}$
Question 5
A bag holds 9 red, 8 blue and 3 green balls. If one ball is taken out without looking, find the probability that it is red.
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A
$\dfrac{1}{3}$
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B
$\dfrac{2}{5}$
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C
$\dfrac{9}{20}$
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D
$\dfrac{9}{11}$
Answer: Option C — with explanation
Total number of balls = 9 + 8 + 3 = 20.
The number of favourable outcomes is the number of red balls, 9.
Probability = $$\dfrac{9}{20}$$.
Common mistakes
- gave the probability of a different colour: $\dfrac{2}{5}$
- assumed the three colours are equally likely: $\dfrac{1}{3}$
- compared the red balls with the other colours instead of with the total: $\dfrac{9}{11}$
Frequently asked questions
Why are there 36 outcomes when two dice are thrown?
Because each die has six faces and the two are independent, so 6 x 6 = 36. Every one of those 36 pairs is equally likely.
What is the difference between at least and exactly?
Exactly three heads means all three coins show heads. At least three heads also means all three, but at least two heads additionally includes the case of three heads, so the two questions give different answers.
Can a probability be greater than 1?
No. A value above 1 means the favourable outcomes outnumber the total, which means one of the two counts is wrong. Always check the answer lies between 0 and 1.
How many face cards are there in a pack?
Twelve, being the jack, queen and king in each of the four suits. The three cards of each suit are the twelve face cards, so the probability of drawing one is 12/52 = 3/13.
