CBSE Class 10 Mathematics Probability Notes Set 07

Basic Terms

An experiment is an activity with well-defined outcomes, such as tossing a coin or throwing a die. A trial is one performance of the experiment. The set of all possible outcomes is the sample space and each outcome is an elementary event. An event is any collection of outcomes.
In classical probability, outcomes are assumed equally likely (a fair coin, an unbiased die, a well-shuffled pack).

Definition and Properties

P(E) = \( \frac{\text{number of outcomes favourable to E}}{\text{total number of possible outcomes}} \)
1. 0 ≤ P(E) ≤ 1.
2. An impossible event has probability 0; a sure event has probability 1.
3. P(E) + P(not E) = 1, so P(not E) = 1 − P(E).
4. The sum of the probabilities of all elementary events of an experiment is 1.
5. For two events that can happen together, P(A or B) = P(A) + P(B) − P(A and B).

Standard Sample Spaces

Coins: one coin has 2 outcomes, two coins 4 (HH, HT, TH, TT), three coins 8 and n coins 2ⁿ.
Dice: one die has 6 outcomes; two dice have 36 ordered pairs. Number of ways to get a sum of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 with two dice: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1.
Playing cards: 52 cards, 4 suits (spades, clubs black; hearts, diamonds red) of 13 cards each. There are 12 face cards (J, Q, K), 4 aces, 4 kings, 4 queens, 4 jacks and 26 red cards.
Leap year: 366 days has 52 weeks and 2 extra days; the probability of 53 Sundays is 2/7. Ordinary year: the probability is 1/7.

Worked Examples

Example 1. A die is thrown once. Find the probability of getting a prime number.
Solution: Prime numbers on a die are 2, 3, 5, so P = 3/6 = 1/2.

Example 2. Two coins are tossed. Find the probability of getting at least one head.
Solution: Outcomes HH, HT, TH, TT. At least one head in 3 of them, so P = 3/4.

Example 3. Two dice are thrown together. Find the probability that the product of the numbers is 12.
Solution: The pairs are (2,6), (6,2), (3,4), (4,3), so P = 4/36 = 1/9.

Example 4. A bag has 3 red and 5 black balls. Two more red balls are added. What is the probability of drawing a red ball?
Solution: Now there are 5 red among 10 balls, so P = 1/2.

Example 5. A card is drawn from a well-shuffled pack. Find the probability that it is neither a king nor a queen.
Solution: Kings and queens together are 8, so P = 44/52 = 11/13.