CBSE Class 9 Mathematics Probability Notes Set 15

Experiment, Sample Space and Event

A random experiment has several possible outcomes and the result of each trial depends on chance. Tossing a coin or rolling a die are examples. The set of all possible outcomes is the sample space S, and each element is a sample point.
Every subset of S is an event. An event with one outcome is simple; one with more is compound. The empty set is the impossible event and S itself is the sure event.

Types of Events

Two events are mutually exclusive when they cannot occur together (their intersection is empty). Events E1, ..., En form a mutually exclusive and exhaustive system if they are pairwise exclusive and their union is S, as the four suits of a pack of cards do.
Two events are independent if the occurrence of one does not affect the other, as with heads on two different coins. Outcomes are equally likely if none is expected in preference to another.

Probability of an Event

For equally likely outcomes, P(E) = n(E) ÷ n(S). Always 0 ≤ P(E) ≤ 1; the impossible event has probability 0 and the sure event has probability 1.
P(not E) = 1 − P(E). For any two events P(A or B) = P(A) + P(B) − P(A and B), and for mutually exclusive events P(A or B) = P(A) + P(B).
Odds in favour of E are n(E) : n(not E), and odds against are n(not E) : n(E).

Standard Sample Spaces

One die: 6 outcomes. Two dice: 36 ordered pairs, with 6 doublets. Three dice: 216 outcomes. Two coins: HH, HT, TH, TT. Three coins: 8 outcomes. A pack of cards: 52 cards in 4 suits of 13, with 12 face cards, 4 aces and 26 red and 26 black cards.
Number of ways to total 2 to 12 with two dice: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.

Worked Examples

Example 1: Three dice thrown together, total at least 6. Total less than 6 happens in 10 of 216 ways, so P = 1 − 10/216 = 103/108.
Example 2: Two dice, sum 4. Favourable = (1,3), (3,1), (2,2), so P = 3/36 = 1/12.
Example 3: Two children from 2 boys and 3 girls. The sample space has 10 pairs; 'both girls' has 3 and 'at least one boy' has 7, and these two events are mutually exclusive.