Class 10 Maths Notes Chapter 14 — Probability#
Key terms#
- Experiment — an action with well-defined outcomes, such as tossing a coin.
- Outcome — a possible result of the experiment.
- Equally likely outcomes — each outcome has the same chance of happening. A fair coin, a fair die and a well-shuffled deck give equally likely outcomes.
- Event — a collection of outcomes, such as “getting an even number”.
- Elementary event — an event with only one outcome.
- Sure (certain) event — always happens. Impossible event — never happens.
Theoretical probability#
$$P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Number of all possible outcomes}}$$
This assumes the outcomes are equally likely.
| Fact |
Value |
| Range |
$0 \leq P(E) \leq 1$ |
| Sure event |
$P = 1$ |
| Impossible event |
$P = 0$ |
| Sum of probabilities of all elementary events |
$1$ |
| Complementary events ($E$ and “not $E$”) |
$P(E) + P(\bar{E}) = 1$ |
- A probability can never be negative or more than 1. So $-1.5$ and $\frac{4}{3}$ can never be probabilities, but $15\%$ and $0.7$ can.
- “Not $E$” questions: find $P(E)$, then $P(\text{not } E) = 1 - P(E)$.
Common experiments#
| Experiment |
Possible outcomes |
Total |
| One coin |
H, T |
2 |
| Two coins |
HH, HT, TH, TT |
4 |
| Three coins |
HHH, HHT, HTH, THH, HTT, THT, TTH, TTT |
8 |
| One die |
1, 2, 3, 4, 5, 6 |
6 |
| Two dice |
(1, 1), (1, 2), …, (6, 6) |
36 |
- With two dice, the sums 2 to 12 are not equally likely. A sum of 7 happens in 6 ways, but a sum of 2 or 12 in only 1 way.
- With two coins, “one head and one tail” happens in 2 ways (HT, TH), so it is twice as likely as two heads.
- On a die: primes are 2, 3, 5; even numbers are 2, 4, 6; odd numbers are 1, 3, 5.
A deck of 52 cards#
|
Count |
| Suits: spades ♠, clubs ♣ (black), hearts ♥, diamonds ♦ (red) |
4 suits × 13 cards |
| Red cards / black cards |
26 each |
| Each suit: A, 2–10, J, Q, K |
13 cards |
| Face cards (J, Q, K) |
12 (3 per suit) |
| Aces / kings / queens / jacks |
4 each |
Geometric probability#
When a point is equally likely to land anywhere in a region:
$$P = \frac{\text{favourable area}}{\text{total area}}$$Common mistakes#
- Treating outcomes as equally likely when they are not, for example “a car starts or does not start”.
- Counting the sums of two dice as 11 equally likely outcomes. List all 36 pairs.
- Forgetting that when a card or item is not replaced, the total for the next draw goes down by one.
- Counting the ace as a face card. Face cards are only J, Q and K.
- Saying “between 2 and 6” includes 2 and 6. It means 3, 4 and 5.