Probability of Simple Events — Print
Probability is a fraction: what you want over everything that could happen — as long as every outcome is equally likely.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Sample space — The set of all possible outcomes.
- Complement — Everything except the event. P(not A) = 1 - P(A).
A bag holds ten marbles: three orange, three red, four grey. Reach in without looking.
What are the chances of pulling out an orange one?
Because you cannot see, every marble is equally likely. That is the sentence that makes the counting work.
Three of the ten are orange, so the chance is 310 — a number between 0 and 1, as every probability must be.
And the chance of not orange is the other seven: 710. Together they make 1, because something must come out.
So simple probability is counting — but only when the outcomes are equally likely. The moment they are not, counting them gives the wrong answer, and that is where most mistakes come from.
The basic formula
P(event) = number of favourable outcomestotal number of outcomes
Rolling a 6-sided die, P(even) = 36 = 12, because 2, 4 and 6 are even out of six faces.
Every probability sits between 0 and 1. Zero means impossible, one means certain. If you ever get 75, you have made an error.
The hidden condition
That fraction only works when every outcome is equally likely. A spinner with unequal wedges breaks it — then you must use area, not a count of regions.
Listing the sample space
When you are unsure, write out every outcome. For two coins: HH, HT, TH, TT — four outcomes, not three.
People often say "two heads, one of each, two tails — so 13 each". That is wrong, because "one of each" happens two ways. P(one of each) = 24 = 12.
The complement shortcut
Sometimes counting what you do not want is far easier.
P(at least one head in 3 flips) has many cases. But the opposite — no heads at all — is a single case, TTT, with probability 18.
So the answer is 1 - 18 = 78.
Watch for "at least"
The phrase "at least one" is almost always a signal to use the complement.
Theoretical and experimental probability
Theoretical probability comes from reasoning: a fair coin gives P(H) = 12.
Experimental probability comes from trials: flip 50 times, get 28 heads, so 2850 = 0.56.
They rarely match exactly, and they do not have to. The more trials you run, the closer experiment usually drifts toward theory — that is the law of large numbers.
Odds versus probability
These are different. If 3 of 10 marbles are red, the probability of red is 310, but the odds in favour are 3 to 7 — favourable to unfavourable, not favourable to total.
Worked examples
A bag of marbles
A bag holds 5 red, 8 blue and 7 green marbles. One is drawn at random. Find P(not blue).
- Total: 5 + 8 + 7 = 20 Everything that could be drawn.
- Not blue means red or green: 5 + 7 = 12 Count the favourable outcomes directly...
- 1220 = 35 ...or use the complement: 1 - 820 = 1220. Same answer.
35
Using probability backwards
A spinner has 12 equal sections. P(win) = 14. How many sections are winners?
- Set up: w12 = 14 Winners over total equals the given probability.
- Cross-multiply: 4w = 12 Or notice 14 of 12.
- w = 3 Three winning sections.
3 sections.
At least one
A fair coin is flipped 4 times. What is the probability of getting at least one tail?
- Opposite event: no tails at all — every flip is heads. "At least one" points to the complement.
- P(HHHH) = 116 There are 24 = 16 equally likely sequences.
- 1 - 116 = 1516 Everything else has at least one tail.
1516
Words to know
- Outcome
- One possible result of an experiment.
- Sample space
- The set of all possible outcomes.
- Event
- A collection of outcomes you care about, like "rolls an even number".
- Complement
- Everything except the event. P(not A) = 1 - P(A).
- Theoretical vs experimental
- What reasoning predicts vs what actually happened when you tried it.
Mistakes to watch for
Counting outcomes that are not equally likely. Flipping two coins, "one head and one tail" is two outcomes (HT and TH), not one — which is why it is twice as likely as two heads.
Giving an answer above 1 or below 0. A probability is a share of the whole, so it cannot exceed 1 or go negative. An answer outside that range means the count went wrong, not that the event is very likely.
Confusing odds with probability. "3 to 7 against" is the same situation as a probability of 310 — odds compare the two outcomes to each other, probability compares one to the total.
Assuming the last result changes the next. A coin has no memory: after four heads, the next flip is still 12. Believing otherwise is common enough to have a name, the gambler's fallacy.
Probability began with a gambling dispute. In 1654 a French nobleman asked Blaise Pascal how to divide the stakes if a game were interrupted part-way.
Pascal wrote to Pierre de Fermat, and their letters that summer invented the subject.
It took another century before anyone thought to use it for anything but dice and cards — insurance, medicine and physics all came later.
Problem 1
A fair 6-sided die is rolled. Find P(rolling a 5).
Worked answer
1/6
P = 16
Problem 2
A fair 6-sided die is rolled. Find P(even).
Worked answer
1/2
Even faces: 2, 4, 6 — that is 3 of them
36 = 12
Problem 3
A bag holds 3 red and 5 blue marbles. Find P(red).
Worked answer
3/8
3 + 5 = 8 marbles
P(red) = 38
Problem 4
For the same bag (3 red, 5 blue), find P(not red).
Worked answer
5/8
1 - 38 = 58
Same as counting the 5 blue marbles directly.
Problem 5
A spinner has 8 equal sections, 3 of them green. Find P(green).
Worked answer
3/8
P = 38
Problem 6
A bag holds 4 red, 6 blue and 10 green marbles. Find P(not blue).
Worked answer
7/10
Total: 4 + 6 + 10 = 20
Not blue: 4 + 10 = 14
1420 = 710
Problem 7
A spinner has 40 equal sections and P(win) = 15. How many sections are winners?
Worked answer
8
w40 = 15
w = 40 ÷ 5 = 8
Problem 8
A number from 1 to 20 is chosen at random. Find P(multiple of 3).
Worked answer
3/10
Multiples of 3 up to 20: 3, 6, 9, 12, 15, 18 — six of them
620 = 310
Problem 9
Two fair coins are flipped. Find P(exactly one head).
Worked answer
1/2
Sample space: HH, HT, TH, TT — four equally likely outcomes
Exactly one head: HT and TH, so two of them
24 = 12 — not 13, because "one of each" happens two ways.
Problem 10
A spinner is spun 60 times and lands on red 21 times. What is the experimental probability of red, as a decimal?
Worked answer
0.35
2160 = 0.35
Problem 11
If P(A) = 0.35, find P(not A).
Worked answer
0.65
1 - 0.35 = 0.65
Problem 12
A bag has only red and blue marbles, 30 in all, and P(red) = 25. How many are blue?
Worked answer
18
Red: 25 × 30 = 12
Blue: 30 - 12 = 18
Problem 13
Three fair coins are flipped. Find P(at least one head).
Worked answer
7/8
The opposite is no heads at all: TTT
There are 23 = 8 equally likely outcomes, so P(TTT) = 18
1 - 18 = 78
Problem 14
A number from 1 to 30 is chosen at random. Find P(multiple of 4 or 5).
Worked answer
2/5
Multiples of 4: 4, 8, 12, 16, 20, 24, 28 — seven
Multiples of 5: 5, 10, 15, 20, 25, 30 — six
20 is in both lists, so subtract it once: 7 + 6 - 1 = 12
1230 = 25
Problem 15
A bag has 5 red and 7 blue marbles. How many red marbles must be added so that P(red) = 12?
Worked answer
2
5 + x12 + x = 12
Cross-multiply: 10 + 2x = 12 + x
x = 2
Check: 7 red and 7 blue ✓
Problem 16
A spinner is half red and a quarter blue; the rest is split evenly between green and yellow. Find P(green).
Worked answer
1/8
Red and blue take 12 + 14 = 34
That leaves 14 for green and yellow
Split evenly: 14 ÷ 2 = 18