Maharashtra State Board · Class 10 · Mathematics · Algebra Chapter 5

Probability — Formula Sheet

Board Formulas

14 formulas

1.Sample Space
$S = {all possible outcomes}, n (S) = number of outcomes$
Example: one die, S = {1, 2, 3, 4, 5, 6}, n(S) = 6. List outcomes in a fixed order so none is missed.
2.Event
$A \subseteq S$
An event is a subset of the sample space. n(A) is the number of outcomes favourable to A.
3.Probability of an Event★
$P (A) = \frac{n ( A )}{n ( S )}$
$P (A)$ : Probability of event A · $n (A)$ : Number of outcomes favourable to A · $n (S)$ : Total number of outcomes in the sample space
Valid when all outcomes are equally likely. Count carefully — most errors are in n(A).
4.Range of Probability★
$0 \leq P (A) \leq 1$
Since 0 ≤ n(A) ≤ n(S). A probability can be written as a fraction, decimal or percentage, but never negative and never more than 1.
5.Sure (Certain) Event
$P (S) = 1$
An event that always happens, e.g. getting a number less than 7 on a die.
6.Impossible Event
$P (\emptyset) = 0$
An event with no favourable outcome (empty set, written { } or φ in the textbook), e.g. getting 7 on an ordinary die.
7.Equally Likely Outcomes
$P (each outcome) = \frac{1}{n ( S )}$
For a fair coin each face has probability 1/2; for a fair die each face has probability 1/6.
8.Sum of Probabilities of All Outcomes
$\sum P (each outcome) = 1$
Useful as a check when you write a table of probabilities.
9.Complement of an Event
$P (A^{'}) = 1 - P (A)$
A' (also written Ā) means 'A does not happen'. P(A) + P(A') = 1. The Std X textbook only names the complement of an event under 'For more information', so counting the outcomes of 'not A' directly is equally acceptable.
10.Tossing Coins
$n (S) = 2^{n}$
For n coins tossed together (or one coin tossed n times). Two coins: S = {HH, HT, TH, TT}; three coins: 8 outcomes.
11.Throwing Dice
$n (S) = 6^{n}$
One die: 6 outcomes; two dice: 36 ordered pairs (1, 1) to (6, 6).
12.Counting Tip: Sum on Two Dice
$Ways to get sum s = 6 - ∣ s - 7∣, 2 \leq s \leq 12$
Not a textbook formula — only a quick check against your list of pairs; in the exam, write the pairs. Sum 7 has the most ways (6); sums 2 and 12 have one way each.
13.Pack of Playing Cards
$n (S) = 52 = 4 \times 13$
4 suits of 13 cards: spades and clubs (black), hearts and diamonds (red). Face cards are J, Q, K: 3 per suit, 12 in all. Aces are not face cards.
14.Two-Digit Numbers from k Different Non-Zero Digits
$n (S) = k (k - 1) (without repetition), n (S) = k^{2} (with repetition)$
Tens place has k choices; without repetition, the units place has k − 1. If 0 is one of the digits, it cannot be in the tens place — list the numbers instead.