Words you need first
- Experiment: an action with results you cannot be sure of before it happens (toss a coin, roll a die).
- Outcome: one possible result (heads; the number 4).
- Equally likely: no outcome has a better chance than another (a fair coin, a fair die).
- Event: a group of outcomes we care about ("an even number" = {2, 4, 6}).
- Favourable outcomes: the outcomes that make the event happen.
Classical definition of probability
If an experiment has n equally likely outcomes and m of them are favourable to event E, then
P(E) = m ÷ n = (number of favourable outcomes) ÷ (total number of outcomes)
This is also called theoretical probability, because we find it by thinking, not by doing the experiment.
Important: it works only when the outcomes are equally likely. "Six or not six" are two outcomes, but they are not equally likely, so P(six) is not 1/2.
Range of probability: from 0 to 1
Because m can never be less than 0 or more than n, 0 ≤ P(E) ≤ 1.
- Impossible event: P = 0 (getting 7 on a die).
- Sure (certain) event: P = 1 (getting a number less than 7 on a die).
- An elementary event has only one outcome. The probabilities of all elementary events of an experiment add up to 1.
Complementary events: P(E) + P(not E) = 1
The event "not E" (written Ē) happens exactly when E does not. Together E and not E cover every outcome, with no overlap. So
P(not E) = 1 − P(E)
This is a shortcut: if P(rain) = 0.7, then P(no rain) = 0.3 without any counting.
Simple problems on single events: coins, dice, cards, bags
One coin: 2 outcomes (H, T).
One die: 6 outcomes (1 to 6). Even: 2, 4, 6. Prime: 2, 3, 5. Odd: 1, 3, 5.
A deck of 52 playing cards: 4 suits (spades ♠ and clubs ♣ are black; hearts ♥ and diamonds ♦ are red), 13 cards in each suit: A, 2–10, J, Q, K. So there are 26 red and 26 black cards, 4 aces, 4 kings, and 12 face cards (J, Q, K of each suit).
Bag of balls or tickets: count each colour or number, then use good ÷ all.
Method for every problem: (1) list or count all outcomes, (2) count the favourable ones, (3) divide, (4) simplify.
Board exam pattern
Probability usually gives a 1-mark MCQ and a 2- or 3-mark question in CBSE Class 10 (the Statistics and Probability unit has 11 marks). Card and bag questions are the most common. Always write the total outcomes and favourable outcomes before the fraction, and simplify the answer.
Try it: predict, then check
Put 3 red and 7 other-coloured slips in a box. Predict P(red) = 3/10 = 0.3. Now draw one slip, note the colour, put it back, and repeat 30 times. Divide your red count by 30. Is it near 0.3? In the 3D (last step) press "Draw ×50" a few times: the more you draw, the closer the experiment comes to the formula.
Check your understanding
1. What is P(getting 7 on one die)? (0, impossible.)
2. If P(E) = 0.25, what is P(not E)? (0.75)
Key formulas and definitions
- P(E) = number of favourable outcomes ÷ total number of outcomes
- 0 ≤ P(E) ≤ 1
- P(impossible event) = 0, P(sure event) = 1
- P(not E) = 1 − P(E)
- Sum of probabilities of all elementary events = 1
- Deck: 52 cards, 4 suits × 13, 26 red, 26 black, 12 face cards, 4 aces
Worked examples
1. A fair coin is tossed once. Find P(heads).
Outcomes: H, T → 2. Favourable: H → 1. P(heads) = 1/2.
2. A die is rolled once. Find the probability of getting a prime number.
Outcomes 1–6 → 6. Primes: 2, 3, 5 → 3. P = 3/6 = 1/2.
3. A bag has 3 red, 5 blue and 2 green balls. One ball is taken at random. Find P(red) and P(not red).
Total = 10. Red = 3. P(red) = 3/10. P(not red) = 1 − 3/10 = 7/10.
4. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (a) a king, (b) a red face card.
(a) Kings = 4, so P = 4/52 = 1/13. (b) Red face cards = J, Q, K of hearts and diamonds = 6, so P = 6/52 = 3/26.
5. The probability that it will rain tomorrow is 0.35. What is the probability that it will not rain?
P(not rain) = 1 − 0.35 = 0.65.
6. Tickets numbered 1 to 40 are in a box. One is drawn. Find the probability that the number is (a) a multiple of 6, (b) a perfect square.
(a) Multiples of 6: 6, 12, 18, 24, 30, 36 → 6. P = 6/40 = 3/20. (b) Squares: 1, 4, 9, 16, 25, 36 → 6. P = 6/40 = 3/20.
7. A bag has some red balls and 12 blue balls. The probability of a red ball is 1/4. How many red balls are there?
Let red = x. x ÷ (x + 12) = 1/4 → 4x = x + 12 → 3x = 12 → x = 4 red balls.
8. What is the probability that a leap year has 53 Sundays?
A leap year has 366 days = 52 weeks + 2 extra days. The 2 extra days can be (Sun, Mon), (Mon, Tue), (Tue, Wed), (Wed, Thu), (Thu, Fri), (Fri, Sat), (Sat, Sun): 7 equally likely pairs. Two of them contain Sunday. P = 2/7.
Common mistakes
- Counting colours instead of balls. 3 red and 5 blue balls give P(red) = 3/8, not 1/2.
- Using good ÷ all when outcomes are not equally likely ("6 or not 6" does not make P(6) = 1/2).
- Giving an answer more than 1 or negative. Every probability must be between 0 and 1.
- Forgetting that face cards are only J, Q, K (12 cards). The ace is not a face card.