Students can go through AP 10th Class Maths Notes Chapter 14 Probability to understand and remember the concepts easily.
Probability 10th Class Notes Maths Chapter 14
→ Probability : The branch of mathematics which measures the chance of the occurrence of an event using numbers is called probability.
The chance that an event will or will not occur is expressed between 0 and 1.
It can also be represented as a percentage, where 0% denotes an impossible event and 100 % implies a certain event.
Probability of an Event E is represented by P(E).
Random experiment: An experiment is called a random experiment if
i) all its possible outcomes are known beforehand and
ii) it is not possible to predict the exact outcome in advance.
→ Outcome is a result of a random experiment. For example, when we roll a dice getting six is an outcome.
→ Event is a set of outcomes. For example, when we roll dice, the probability of getting a number less than five is an event.
An event can have a single outcome.
→ Experimental Probability : Experimental probability can be applied to any event associated with an experiment that is repeated a large number of times.
A trial is when the experiment is performed once. It is also known as empirical probability.
→ Experimental or empirical probability : P(E) = Number of trials where the event occurred/Total Number of Trials
→ Theoretical Probability : Theoretical Probability, P(E) = Number of Outcomes Favourable to E / Number of all possible outcomes of the experiment.
Here we assume that the outcomes of the experiment are equally likely.
→ Elementary Event: An event having only one outcome of the experiment is called an elementary event.
→ Impossible Event: An event that has no chance of occurring is called an Impossible event, i.e. P(E) = 0.
→ Sure Event : An event that has1 a 100% probability of occurrence is called a sure event. The probability of occurrence of a sure event is one.
→ Range of Probability of an Event: Probability can range between 0 and 1. where 0 probability means the event to be an impossible one and probability of 1 indicates a certain event i.e. 0 ≤ P (E) ≤ 1.
→ Types of Events: Events can be classified into sure events, impossible events normal events, complementary events, and elementary events.
- Sure Events : A sure event is am event that will always occur, irrespective of the other factors.
- Example : The probability of getting a number between 1 and 6, when a dice is rolled is 1. Hence, the probability of a sure event is always equal to 1.
- Impossible Events: An impossible event is an event that can never happen.
- Example : The probability of getting a number greater than 6, when a dice is rolled is 0. Hence, the probability of an impossible event is always equal to 0.
- Normal Events : A normal event is an event, that can have any probability, i.e. between 0 and 1 inclusive.
A sure event and an impossible event, are also normal events. For example, the probability of getting 1 on a die is 1/6. - Complementary Events: A complementary event is an event, such that the sum of the probability of occurring an event and the probability of not occurring the same event is equal to 1.
- Example : The probability of getting a number less than 5 on a die is, 4/6, and then the probability of not getting a number less than 5 is 2/6.
Hence, the sum of their probabilities is equal to 1.
Complementary events are two outcomes of an event that are the only two possible outcomes. This is like flipping a coin and getting heads or tails.
P(E) + P(\(\overline{\mathbf{E}}\)) = 1
where E and \(\overline{\mathbf{E}}\) are complementary events.
→ Two or more events which have an equal probability of occurrence are said to be equally likely events.
→ Non-Equally likely outcome : A non-equaily likely outcome is an outcome, that has I uncertainty in its outcomes and also depends on the various other factors. For example, starting a car, and the car starts or does not start, this gives a non-equally likely outcome, because the car will always start unless it needs maintenance/service.
→ Cards Probability : A deck/pack of playing cards consists of 52 cards which are separated into four suits of hearts (13) , diamonds (13) , clubs (13), and spades (13) and in to two colours red (26) and black (26).
The symbols that are used to represent the four suits are:
- Hearts (13) : A red heart symbol (♥) represents the heart suit.
- Diamonds (13) : A red diamond symbol (♦) represents the diamond suit.
- Clubs (13) : A black club symbol (♣) represents the club suit.
- Spades (13) : A black spade symbol (♠) represents the spade suit
Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King are the 13 cards that make up each suit. Jack, Queen and King are called face cards.
When a coin is tossed the possible outcomes are {Head, Tail}. Total 2. ,
When two coins are tossed simultaneously or a coin is tossed for two times the total ! outcomes are 22 = 4.
When three coins are tossed simultaneously or a coin is tossed for three times the total outcomes are 23 = 8.
When n-coins are tossed simultaneously or a coin is tossed for n-times the total outcomes are 2n.
When a dice is rolled the possible outcomes are {1,2, 3, 4, 5 and 6}. Total 6.
Similarly, when two dice are rolled or a dice is rolled for two times the total outcomes are 62 = 36.
When three dice are rolled or a dice is rolled for three times the total outcomes are 63 = 216.
When n-dice are rolled or a dice is rolled for n-times the total outcomes are 6n = 216.
When a card is drawn at random from a deck of cards, the possible outcomes are {Ace, 2, 3, 4, 5, 6, 7, 8, 9,10, Jack, Queen, and King} × 4. Total 52.
→ Types of Probability :
- Experimental Probability
- Theoretical Probability
p(A) = \(\frac{\text { Number of favourable outcomes }}{\text { Number of all possible outcomes }}\) = \(\frac{m}{m+n}\)
and the probability of not happening of A, denoted by P(\(\overline{\mathbf{A}}\)) is given by
p(\(\overline{\mathbf{E}}\)) = \(\frac{\text { Number of unfavourable outcomes }}{\text { Number of all possible outcomes }}\) = \(\frac{n}{m+n}\)
→ Mind map and formulae :
