CBSE Class 10 Probability Important Formulas and concepts for exams

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CLASS X : CHAPTER - 15

PROBABILITY

IMPORTANT FORMULAS & CONCEPTS

PROBABILITY

Experimental or empirical probability P(E) of an event E is defined as:
P(E) = \(\frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}\)
The theoretical probability (also called classical probability) of an event A, written as P(A), is defined as:
P(A) = \(\frac{\text{Number of outcomes favourable to A}}{\text{Number of all possible outcomes of the experiment}}\)
Two or more events of an experiment, where occurrence of an event prevents occurrences of all other events, are called Mutually Exclusive Events.

Answer:
1. Understand the difference between empirical and theoretical probability.
2. Empirical probability is based on actual experiments and observations.
3. Theoretical probability is based on equally likely outcomes without actually performing the experiment.

Teacher's Note:
a) Core concepts include fundamental definitions of probability.
b) Students often confuse experimental trials with theoretical sample space outcomes.
c) Remember that experimental probability approaches theoretical probability as the number of trials increases.

COMPLIMENTARY EVENTS AND PROBABILITY

We denote the event 'not E' by Ē. This is called the complement event of event E.
So, P(E) + P(not E) = 1
i.e., P(E) + P(Ē) = 1, which gives us P(Ē) = 1 - P(E).
In general, it is true that for an event E, P(Ē) = 1 - P(E).
- The probability of an event which is impossible to occur is 0. Such an event is called an impossible event.
- The probability of an event which is sure (or certain) to occur is 1. Such an event is called a sure event or a certain event.
- The probability of an event E is a number P(E) such that 0 ≤ P(E) ≤ 1.
- An event having only one outcome is called an elementary event. The sum of the probabilities of all the elementary events of an experiment is 1.

Answer:
1. Complementary events cover all remaining possibilities of an event not occurring.
2. The sum of probabilities of an event and its complement is always equal to 1.
3. Probability values always lie strictly between 0 and 1 inclusive.

Teacher's Note:
a) Master the formula P(E) + P(not E) = 1 for quick problem solving.
b) Never write a probability value greater than 1 or less than 0.
c) Elementary events sum up to exactly 1 in any given random experiment.

DECK OF CARDS AND PROBABILITY

A deck of playing cards consists of 52 cards which are divided into 4 suits of 13 cards each.
They are black spades (♠), red hearts (♥), red diamonds (♦) and black clubs (♣).
The cards in each suit are Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2. Kings, Queens and Jacks are called face cards.

Answer:
1. Total number of cards in a standard deck is 52.
2. There are 26 red cards and 26 black cards in total.
3. There are 12 face cards in total, with 3 face cards in each of the 4 suits.

Teacher's Note:
a) Memorize the breakdown of suits, colors, and face cards to solve card-based probability problems efficiently.
b) Do not count Aces as face cards; face cards are strictly Kings, Queens, and Jacks.
c) Each suit contains exactly 13 cards comprising numbers 2 to 10 and 4 honor cards (Ace, Jack, Queen, King).

1. Define equally likely events. [1 Mark]

Answer:
1. Two or more events are said to be equally likely if each one of them has an equal chance of occurrence.
2. Example includes tossing a fair coin, where heads and tails have equal chances.

Teacher's Note:
a) This is a fundamental definition in classical probability.
b) It assumes symmetry and fairness in the random experiment.

2. Define mutually exclusive events. [1 Mark]

Answer:
1. Two or more events are mutually exclusive if the occurrence of each event prevents the every other event.
2. In such cases, no two events can occur simultaneously.

Teacher's Note:
a) Mutually exclusive events have an intersection of empty sets.
b) The probability of the union of mutually exclusive events is the sum of their individual probabilities.

3. Define complementary events. [1 Mark]

Answer:
1. Consider an event has few outcomes.
2. Event of all other outcomes in the sample survey which are not in the favourable event is called Complementary event.

Teacher's Note:
a) The sum of the probability of an event and its complement is always equal to 1.
b) Written mathematically as P(E) + P(not E) = 1.

4. Define exhaustive events. [1 Mark]

Answer:
1. All the events are exhaustive events if their union is the sample space.
2. This means that the set of events covers all possible outcomes of the experiment.

Teacher's Note:
a) Exhaustive events ensure that no potential outcome is left out.
b) Every sample space has at least one set of exhaustive events.

5. Define sure events. [1 Mark]

Answer:
1. The sample space of a random experiment is called sure or certain event.
2. Any one of its elements will surely occur in any trial of the experiment.

Teacher's Note:
a) The probability of a sure event is always exactly equal to 1.
b) It represents certainty in the context of probability theory.

6. Define impossible event. [1 Mark]

Answer:
1. An event which will occur on any account is called an impossible event.
2. Such an event contains no outcomes from the sample space (empty set).

Teacher's Note:
a) The probability of an impossible event is always equal to 0.
b) Note the phrasing in the original text indicating events that cannot happen under any circumstances.