The probability scale
"It will surely rain" or "a snowfall in Mumbai" describe how likely events are. A number from 0 to 1 says it more exactly.
The probability of an event is a number from 0 to 1. 0 means impossible, 1 means certain, ½ means equally likely to happen or not. Closer to 1 is more likely.
Formula: 0 ≤ P(E) ≤ 1
In the figure: An impossible event sits at 0, a certain one at 1, an even chance at 1/2.
Rule 1: Place on the scale — Decide impossible, unlikely, even, likely or certain, then give a number.
- Easy: The sun rises tomorrow: about 1.
- Medium: Getting heads on a fair coin: 1/2.
- Exam-type: A die shows 7: 0.
Rule 2: Compare events — The event with the larger probability is more likely.
- Easy: P = 0.8 is more likely than P = 0.3.
- Medium: P = 1/4 is less likely than P = 1/2.
- Exam-type: A probability of 1.2 or −0.3 is impossible to have.
Probability is never negative and never more than 1.
Saying a probability of 2 or 150% for an event.
"Zero never, one sure, half fifty-fifty."
Practice questions
- 1A person born today lives for 500 years. Is this event certain, impossible or neither?
- 2The sun rises in the east tomorrow. Is this event certain, impossible or neither?
- 3Four events have probabilities 1/10, 1/3, 1/5, 1/4. Which event is the most likely?
- 4Four events have probabilities 1/10, 9/10, 1/3, 4/5. Which event is the most likely?
- 5An event has probability 0. Which description fits it best?
- 6Bag A has 5 red balls out of 7 balls. Bag B has 5 red balls out of 12 balls. From which bag is a red ball more likely to be drawn at random?
- 7Bag A has 4 red balls out of 9 balls. Bag B has 2 red balls out of 8 balls. From which bag is a red ball more likely to be drawn at random?
- 8The probability of an event is 1/5. If the experiment is repeated 100 times, about how many times do we expect the event to occur?
Show answers (try them yourself first)
- 1impossible
- 2certain
- 31/3
- 49/10
- 5impossible
- 6Bag A
- 7Bag A
- 820
Experimental probability
To find how often a drawing pin lands point up, you drop it many times and count. No formula can tell you that.
Experimental probability = (number of times the event happened) ÷ (total number of trials). More trials give a more reliable value.
Formula: P(E) = times E occurred ÷ total trials
In the figure: 27 heads in 50 tosses gives experimental P(heads) = 27/50 = 0.54.
Rule 1: Compute it — Divide the count of the event by the total trials.
- Easy: Rain on 12 of 30 days: 12/30 = 2/5.
- Medium: Hit the target 8 of 20 shots: 0.4.
- Exam-type: A defective bulb in 5 of 200 bulbs: 1/40.
Rule 2: Use it to predict — Multiply the experimental probability by a larger number of trials.
- Easy: P = 0.05 defective in 1000 items: about 50.
- Medium: P(six) = 1/6 in 60 throws: about 10 sixes.
- Exam-type: Probabilities of all outcomes of one experiment add to 1.
Results vary from one set of trials to another, but settle near one value for many trials.
Dividing by the number of favourable outcomes instead of total trials.
"Count the hits, divide by tries."
Practice questions
- 1A coin is tossed 20 times and shows heads 19 times. What is the experimental probability of heads?
- 2A bag was shaken and a bead drawn and replaced 33 times. The results: Red 5, Blue 10, Green 5, Yellow 13. What is the experimental probability of drawing Red?
- 3A bag was shaken and a bead drawn and replaced 40 times. The results: Red 4, Blue 15, Green 6, Yellow 15. What is the experimental probability of drawing Blue?
- 4A die is thrown 101 times. Outcomes 1 to 6 occurred 19, 11, 14, 18, 17, 22 times respectively. Find the experimental probability of getting an odd number.
- 5In a sample of 50 bulbs, 6 were found defective. Based on this, about how many defective bulbs would you expect in a lot of 500 bulbs?
- 6In 25 throws of a die, the number 6 appeared 5 times. What is the experimental probability of getting a 6?
- 7In 20 tosses of a coin, heads appeared 8 times. Then the coin is tossed 10 more times and heads appears every time. What is the experimental probability of heads now?
- 8Marks of 43 students were grouped as follows: 0–20: 5, 20–40: 9, 40–60: 8, 60–80: 11, 80–100: 10 students. If a student is chosen at random, what is the experimental probability that the student scored 60 or more?
Show answers (try them yourself first)
- 119/20
- 25/33
- 33/8
- 450/101
- 560
- 61/5
- 73/5
- 821/43
Experimental probability
In the app you can move this figure with your finger.
Open in the appTheoretical probability and sample space
A fair die has six faces, all equally likely. You can predict the chance of an even number without rolling.
When all outcomes are equally likely, P(E) = favourable outcomes ÷ total outcomes. The list of all outcomes is the sample space.
Formula: P(E) = n(E) ÷ n(S)
In the figure: Sample space {1, 2, 3, 4, 5, 6}; the even numbers 2, 4, 6 are 3 of 6.
Rule 1: Dice and coins — List the sample space, count favourable outcomes.
- Easy: P(5 on a die) = 1/6.
- Medium: P(number > 4) = 2/6 = 1/3.
- Exam-type: P(a prime on a die) = 3/6 = 1/2.
Rule 2: Cards and bags — Total outcomes are the total items in the bag or pack.
- Easy: Bag: 3 red, 5 blue. P(red) = 3/8.
- Medium: 52 cards: P(a king) = 4/52 = 1/13.
- Exam-type: Letters of MATHS: P(vowel) = 1/5.
The formula works only if the outcomes are equally likely.
Counting outcomes that are not equally likely, e.g. "rain or no rain = 1/2".
"Wanted over all."
Practice questions
- 1A fair die is thrown once. Find the probability of getting a number greater than 4.
- 2A bag contains 5 green beads and 3 yellow beads. One is drawn at random. What is the probability that it is green?
- 3How many outcomes are in the sample space when a coin is tossed three times?
- 4A bag has 4 red, 6 blue and 4 green balls. One ball is drawn at random. What is the probability that it is red or green?
- 5A spinner has 15 equal sectors numbered 1 to 15. What is the probability that it stops on a multiple of 2?
- 6Cards are numbered 1 to 30, well shuffled, and one is drawn. Find the probability that the number is a prime number.
- 7Two fair dice are thrown together. What is the probability that the sum of the numbers is 6?
- 8A bag has 5 red balls out of 12 balls. If 4 more red balls are added, what is the new probability of drawing a red ball?
Show answers (try them yourself first)
- 11/3
- 25/8
- 38
- 44/7
- 57/15
- 61/3
- 75/36
- 89/16
Theoretical probability and sample space
In the app you can move this figure with your finger.
Open in the appComplementary and mutually exclusive events
If the chance of winning a match is 0.7, the chance of not winning is 0.3, because win or not-win covers everything.
The complement of E (not E) has P(not E) = 1 − P(E). Events that cannot happen together are mutually exclusive, and then P(A or B) = P(A) + P(B).
Formula: P(not E) = 1 − P(E); P(A or B) = P(A) + P(B)
In the figure: The two bars add up to 1: 0.7 + 0.3.
Rule 1: Use the complement — "At least one" or "not" is easiest as 1 minus the opposite.
- Easy: P(win) = 0.65 ⇒ P(not win) = 0.35.
- Medium: P(no 6 on a die) = 1 − 1/6 = 5/6.
- Exam-type: P(not a king) = 1 − 1/13 = 12/13.
Rule 2: Add for "or" — Add only when the events cannot happen together.
- Easy: Die: P(1 or 2) = 1/6 + 1/6 = 1/3.
- Medium: Bag R 0.2, B 0.5: P(R or B) = 0.7.
- Exam-type: Even and odd on a die are mutually exclusive: 1/2 + 1/2 = 1.
Complementary events are always mutually exclusive, but mutually exclusive events need not be complementary.
Adding probabilities of events that can happen together, like "even" and "greater than 3".
"Not = one minus."
Practice questions
- 1A and B are mutually exclusive events with P(A) = 3/10 and P(B) = 1/10. Find P(A or B).
- 2A bag has 4 red, 7 blue and 6 green balls. One ball is drawn at random. Find the probability that it is red or blue.
- 3A bag has 3 red, 6 blue and 6 green balls. One ball is drawn at random. Find the probability that it is red or blue.
- 4A and B are mutually exclusive events with P(A) = 3/10 and P(B) = 2/5. Find the probability that neither A nor B happens.
- 5A and B are mutually exclusive events with P(A) = 1/6 and P(B) = 1/3. Find the probability that neither A nor B happens.
- 6Two dice are thrown. What is the probability that the sum is less than 5?
- 7Two dice are thrown. What is the probability that the sum is less than 4?
- 8A, B and C are the only possible outcomes of an experiment and no two can happen together. If P(A) = 3/10 and P(B) = 9/20, find P(C).
Show answers (try them yourself first)
- 12/5
- 211/17
- 33/5
- 43/10
- 51/2
- 61/6
- 71/12
- 81/4
Tree diagrams and compound experiments
Tossing two coins one after another, each toss branches into H or T. Drawing the branches shows no outcome is missed.
A tree diagram lists every outcome of a multi-step experiment. Number of outcomes = product of the choices at each step (2 × 2 = 4). Probability of one path = product along the branches.
Formula: Outcomes = n₁ × n₂; P(path) = product of branch probabilities
In the figure: Two coin tosses give HH, HT, TH, TT: four equally likely outcomes.
Rule 1: List and count — Draw branches step by step, then count the favourable paths.
- Easy: Two coins: P(exactly one head) = 2/4 = 1/2.
- Medium: Three coins: 8 outcomes; P(HHH) = 1/8.
- Exam-type: Die and coin: 6 × 2 = 12 outcomes.
Rule 2: Multiply along branches — For independent steps multiply the probabilities of each step.
- Easy: P(H then H) = 1/2 × 1/2 = 1/4.
- Medium: P(6 then 6) on two throws = 1/36.
- Exam-type: P(at least one head in 2 tosses) = 1 − 1/4 = 3/4.
Add the probabilities of different paths that give the wanted result.
Counting HT and TH as one outcome.
"Multiply along, add across."
Practice questions
- 1A coin is tossed 3 times. How many different outcomes (paths in the tree diagram) are there?
- 2A spinner with 4 equal parts and a coin are used together. How many outcomes does the tree diagram have?
- 3A fair coin is tossed 4 times. What is the probability of getting exactly 1 head?
- 4A bag has 4 red and 5 blue balls. A ball is drawn, noted and put back; then a second ball is drawn. Find the probability that both are red.
- 5A fair coin is tossed 5 times. Find the probability of getting at least one head.
- 6A coin is tossed and a die is thrown. Find the probability of getting a head and an odd number on the die.
- 7A bag has 3 red and 3 blue balls. Two balls are drawn one after the other without replacement. Find the probability that both are red.
- 8A bag has 3 red and 2 blue balls. Two balls are drawn one after the other without replacement. Find the probability that exactly one is red.
Show answers (try them yourself first)
- 18
- 28
- 31/4
- 416/81
- 531/32
- 61/4
- 71/5
- 83/5