Probability
Classical probability is just counting: how many outcomes match what you want, out of all the equally likely outcomes possible.
1. The Classical Definition
P(E) always lies between 0 (impossible) and 1 (certain), inclusive.
Favourable outcomes: {5, 6} → 2 outcomes. Total outcomes: {1,2,3,4,5,6} → 6.
P = 2/6 = 1/3
Favourable pairs: (2,6),(3,5),(4,4),(5,3),(6,2) → 5 outcomes. Total outcomes (6×6) = 36.
P = 5/36
Kings + queens = 4 + 4 = 8 unfavourable cards. P(king or queen) = 8/52.
P(neither) = 1 − 8/52 = 44/52 = 11/13
A leap year has 366 days = 52 complete weeks + 2 extra days. These 2 extra days can be any of 7 equally likely consecutive pairs: (Sun,Mon), (Mon,Tue), …, (Sat,Sun). Exactly 2 of these 7 pairs contain a Sunday.
P = 2/7
- Forgetting that “neither A nor B” means 1 − P(A or B), and being careful not to double-count when A and B overlap.
- For two-dice problems, listing sums instead of ordered pairs — (2,6) and (6,2) are different outcomes and both count.
- Forgetting that a standard deck has 4 suits × 13 cards = 52 total, with 4 of each rank (ace through king).
Key Vocabulary Recap
Syllabus Reference
Practice questions
Probability — Practice
Covers: the classical definition of probability applied to dice, cards, coins, and everyday situations. 10 questions.
A die is thrown once. Find the probability of getting a number greater than 4.
A card is drawn from a well-shuffled deck of 52 cards. Find the probability that it is a king.
Two coins are tossed simultaneously. Find the probability of getting at least one head.
A bag contains 3 red and 5 black balls. A ball is drawn at random. Find the probability that it is red.
Find the probability that a randomly selected leap year has 53 Sundays.
A box contains 90 discs numbered 1 to 90. If one disc is drawn at random, find the probability that it bears a perfect square number.
Two dice are thrown simultaneously. Find the probability of getting a sum of 8.
From a pack of 52 cards, one card is drawn. Find the probability that it is neither a king nor a queen.
A bag contains 4 red, 5 blue, and 3 green balls. A ball is drawn at random. Find the probability that it is not blue.
Three unbiased coins are tossed together. Find the probability of getting exactly two heads.
More Practice
A bag contains 5 red, 8 white, and 7 black balls. A ball is drawn at random. Find the probability that it is not black.
A number is selected at random from 1 to 25. Find the probability that it is a prime number.
Two dice are thrown together. Find the probability that the product of the numbers is 12.
A letter is chosen at random from the word “MATHEMATICS.” Find the probability that it is a vowel.
A coin is tossed 3 times. Hanif wins if all three tosses give the same result. Find the probability that Hanif loses.
A box of 800 electric bulbs contains 36 defective ones. One bulb is drawn at random. Find the probability that it is non-defective.
Two customers are each equally likely to visit a shop on any of 5 days (Tuesday to Saturday). Find the probability that both visit on the same day.
A die is thrown twice. Find the probability that 5 does not come up either time.
12 defective pens are accidentally mixed with 132 good ones. One pen is drawn at random. Find the probability that it is good.
A jar contains 24 marbles, some green and some blue. If the probability of drawing a green marble is 2/3, find the number of blue marbles.
Worked solutions
Show worked solutions
Probability — Practice: Solutions
Die, P(>4).
Card is a king.
Two coins, at least one head.
3 red, 5 black balls. P(red).
Leap year, P(53 Sundays).
Discs 1-90, P(perfect square).
Two dice, P(sum=8).
P(neither king nor queen).
4 red,5 blue,3 green. P(not blue).
Three coins, P(exactly 2 heads).
More Practice — Solutions
5 red,8 white,7 black. P(not black).
P(prime) from 1-25.
Two dice, P(product=12).
MATHEMATICS, P(vowel).
3 coin tosses, P(Hanif loses).
800 bulbs, 36 defective. P(non-defective).
Two customers, 5 days each. P(same day).
Die thrown twice, P(no 5 either time).
12 defective, 132 good pens. P(good).
24 marbles, P(green)=2/3. Find blue.
Book a free trial class
Tell us a little about your requirement.