CBSE Class 10 Mathematics Probability Notes Set 04

Introduction

Probability is defined as the quantitative expression of the likelihood of the happening of an event. If an event has a low probability, then it is unlikely to happen. If an event has a high probability, then it is likely to happen. Probability is mainly of two types:
1. Theoretical probability
2. Experimental probability

In theoretical probability, we predict the results without actually performing the experiment. The theoretical (classical) probability of an event E, written as P(E), is defined as

\[ P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Total number of outcomes}} \]

For example, the probability of getting a head when a coin is tossed once is \( \frac{1}{2} \).

The experimental or empirical probability of an event F, written as P(F), is based on the results of actual experiments. It is defined as

\[ P(F) = \frac{\text{Number of times event } F \text{ occurred}}{\text{Total number of times the experiment was performed}} \]

For example, suppose a coin is tossed 100 times in which a head is obtained 55 times and a tail 45 times. Then the probability of getting a head is \( P(H) = \frac{55}{100} = \frac{11}{20} \).

Note. As the number of trials in an experiment increases, we may expect the experimental and theoretical probabilities to be nearly the same.

Some Basic Terms

Random experiment. An experiment is said to be a random experiment if it has more than one possible outcome and its outcome cannot be predicted or determined in advance.
Outcome. A possible result of a random experiment is called its outcome. For example, if the experiment consists of tossing a coin twice, its outcomes are HH, HT, TH and TT.
Sample space. The set of all possible outcomes of a random experiment is called its sample space, denoted by S. For example, tossing a coin twice gives \( S = \{HH, HT, TH, TT\} \).
Event. A collection of some outcomes of a random experiment (a subset of the sample space) is called an event, denoted by E. For example, if a coin is tossed twice and E is the event of getting at least one head, then E contains HT, HH and TH.

Types of Events

1. Elementary event. An event with only one outcome. For example, when one card is drawn from a well-shuffled deck, drawing the queen of hearts is an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.
2. Compound event. An event which has more than one elementary event. For example, if we toss two coins, getting at least one head is a compound event, as it contains three elementary events HH, HT and TH.
3. Equally likely events. Events are equally likely if the chances of happening of all the outcomes of an experiment are equal. For example, when a die is rolled, getting 1, 2 or any other number are equally likely.
4. Complementary events. The complement of an event E is the event 'not E', written \( \overline{E} \). For example, if E is the event 'getting a head' on tossing a coin, then \( \overline{E} \) is getting a tail. Then \( P(E) = \frac{1}{2} \) and \( P(\overline{E}) = \frac{1}{2} \), so \( P(E) + P(\overline{E}) = 1 \) or \( P(\overline{E}) = 1 - P(E) \). This is true for any event E.

Note.
1. The probability of an impossible event (one that cannot occur) is 0.
2. The probability of a sure or certain event is 1.
So the probability of any event always lies between 0 and 1: \( 0 \leq P(E) \leq 1 \).

Sum of Elementary Events

If an event E can happen in any one of n different exclusive ways with probabilities \( p_1, p_2, p_3, \ldots, p_n \) respectively, then the probability of the event E is the sum of the individual probabilities \( p_1 + p_2 + \cdots + p_n \). For example, when a die is rolled, the number of even outcomes (2, 4 or 6) is 3 out of 6, so P(even number) \( = \frac{3}{6} = \frac{1}{2} \). Also P(2) = P(4) = P(6) = \( \frac{1}{6} \), and \( \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2} \).

Pack of Playing Cards

1. A deck (pack) of cards contains 52 cards, of which 26 are red and 26 are black.
2. There are four suits, each containing 13 cards.

Colour	Suit	Suit
Red cards (26)	Diamonds (13)	Hearts (13)
Black cards (26)	Spades (13)	Clubs (13)

3. The cards in each suit are ace (A), king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2.
4. Kings, queens and jacks are called face cards (4 + 4 + 4 = 12).

Solved Examples

Example 1. What is the probability of getting a number 9 in a single throw of a die?
Answer:
The possible outcomes are 1, 2, 3, 4, 5, 6. Let E be the event of getting a number 9 in a single throw.
\( P(E) = \frac{0}{6} = 0 \). The event E is an impossible event.

Example 2. A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that the card drawn is (i) a black card (ii) a non-king (iii) a king or a jack (iv) neither an ace nor a jack.
Answer:
Total number of possible outcomes = 52.
(i) Black cards = 26. \( P = \frac{26}{52} = \frac{1}{2} \).
(ii) There are 4 kings, so the number of non-kings = 48. \( P = \frac{48}{52} = \frac{12}{13} \).
(iii) There are 4 kings and 4 jacks, so 8 favourable cards. \( P = \frac{8}{52} = \frac{2}{13} \).
(iv) There are 4 aces and 4 jacks. \( P(\text{neither an ace nor a jack}) = 1 - \frac{4 + 4}{52} = 1 - \frac{8}{52} = \frac{44}{52} = \frac{11}{13} \).

Example 3. Three unbiased coins are tossed together. Find the probability of getting (i) one head (ii) two heads (iii) all heads (iv) at least two heads.
Answer:
All possible outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Total = 8.
(i) One head: HTT, THT, TTH. \( P = \frac{3}{8} \).
(ii) Two heads: HHT, HTH, THH. \( P = \frac{3}{8} \).
(iii) All heads: HHH. \( P = \frac{1}{8} \).
(iv) At least two heads: HHT, HTH, THH, HHH. \( P = \frac{4}{8} = \frac{1}{2} \).

Example 4. 17 cards numbered 1, 2, 3, ..., 17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability that the number on the card is (i) odd (ii) a prime (iii) divisible by 3 (iv) divisible by 3 and 2 both.
Answer:
Total number of elementary events = 17.
(i) Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, that is 9 cards. \( P = \frac{9}{17} \).
(ii) Prime numbers: 2, 3, 5, 7, 11, 13, 17, that is 7 cards. \( P = \frac{7}{17} \).
(iii) Divisible by 3: 3, 6, 9, 12, 15, that is 5 cards. \( P = \frac{5}{17} \).
(iv) Divisible by both 3 and 2 means a multiple of 6: 6 and 12, that is 2 cards. \( P = \frac{2}{17} \).

Summary

1. Probability is the quantitative expression of the likelihood of the happening of an event.
2. Theoretical probability: \( P(E) = \frac{\text{Number of outcomes favourable to } E}{\text{Total number of outcomes}} = \frac{n(E)}{n(S)} \).
3. Experimental (empirical) probability: \( P(F) = \frac{\text{Number of times event } F \text{ occurred}}{\text{Total number of times the experiment was performed}} \).
4. The collection of all or some outcomes of an experiment is called an event. An event with only one outcome is an elementary event; an event with more than one elementary event is a compound event.
5. Events whose outcomes have equal chances of happening are equally likely events.
6. Two events are complementary if \( P(E) + P(\overline{E}) = 1 \).
7. The probability of an impossible event is 0 and of a sure event is 1.
8. The sum of the probabilities of all the elementary events of an experiment is 1.
9. For any event E, \( 0 \leq P(E) \leq 1 \).