Probability is a crucial concept in mathematics that helps us make sense of uncertainty and randomness in various real-world situations. Whether you are tossing a coin, rolling a die or predicting the weather, probability plays a key role in analysing and predicting outcomes. In this chapter, we will explore the basics of probability and its significance.
Probability is a way of measuring the chances of something happening. It is measured by the ratio of the favourable events to the whole number of possible events.
When provided with a sample space (S) and an event (E), the probability of an event E occurring is denoted by P(E).
NOTE:
1) The sum of the probabilities of all elementary events in an experiment is 1. (The sum of probability of occurrence of an event and the probability of occurrence of the complement of that event is always 1.)
P(E) + P( E′) = 1
2) Probability is always between 0 and 1.
0 ≤ P(E) ≤ 1
3) If it can't happen at all, then the probability is 0. (When the occurrence of an event is 0, the event is impossible.)
P(E) = 0
4) If we are sure something will happen, the probability is 1. (When the occurrence of an event is 1, the event is certain.)
P(E) = 1
An experiment is any activity associated with a certain result.
For example:
(i) Tossing a coin.
(ii) Throwing a die.
(iii) Selecting a card.
A random experiment is like a test or trial where we can't be sure what will happen. A trial is an action that will result in one or several outcomes.
For example:
(i) Flipping a coin will result in two outcomes.
(ii) Rolling a fair six-sided die will result in six outcomes.
An outcome refers to a possible result or consequence of a random experiment.
For example:
(i) Flipping a coin will result in two outcomes - Heads and tails.
(ii) Rolling a six-sided dice is a random experiment. The possible outcomes for this experiment are the numbers 1 through 6.
If we consider the event of rolling a number 3, then the outcome “3” is one of the possible outcomes of this experiment.
The sample space is all the possible outcomes we could get from a random experiment. It is denoted by S.
For example:
(i) If a coin is tossed, then the sample space would be either getting heads (H) or tails (T).
(ii) If we roll a die, then the sample space would be the numbers 1, 2, 3, 4, 5 or 6.
An event for an experiment is the collection of some outcomes of the random experiment. It is denoted by E.
For example:
(i) Getting a head on tossing a coin.
(ii) Getting a face card when a card is drawn from a pack of 52 cards.
A complementary event for an experiment represents the non-occurrence of the outcomes with respect to sample space. If event E is the occurrence of a particular outcome, then the complementary event is denoted as E'.
For example:
Consider the tossing of a fair six-sided dice. Total outcomes like 1, 2, 3, 5 and 6.
(i) If E is the event of rolling a dice to get 4, then the complementary event E′ would be the event of not rolling a dice to get 4.
(ii) If E is the event of rolling a dice to get 5, then the complementary event E′ would be the event of not rolling a dice to get 5.
When tossing a fair coin, there are two possible outcomes: it can land either heads (H) or tails (T).
When tossing two coins simultaneously, there are more possible outcomes compared to a single coin toss.
The outcomes can be summarised where H represents heads and T represents tails:
When throwing a fair six-sided dice, there are six possible outcomes. Each corresponds to one of the six faces of the dice.
The numbers on the faces are 1, 2, 3, 4, 5 and 6.
NOTE:
→ At least: At least signifies a minimum requirement or a baseline that should be met.
When a dice rolls, "at least" means you are considering the minimum number of successful outcomes or the lower end of a range.
For example: If you roll a dice and are asked for the probability of rolling at least 3, then you need to consider all the outcomes that are 3 and higher values (4, 5 and 6).
Favourable Outcomes = 3, 4, 5, 6
→ At most: At most, it signifies the maximum allowable requirement.
When a dice rolls, "at most" means you are considering the maximum number of successful outcomes or the upper end of a range.
For example: If you roll the same dice and are asked for the probability of rolling at most 3, you need to consider all the outcomes that are 3 and lower values (1 and 2).
Favourable Outcomes = 1, 2, 3
When rolling two six-sided dice simultaneously, there are multiple outcomes and the possible combinations of the two dice determine the results. Each dice has six faces with numbers 1 to 6.
There are a total of 36 outcomes and the sample space is as follows:
Shuffling a deck of playing cards is a common practice to randomise the order of the cards.
A pack of playing cards consists of 52 cards which are divided into 4 suits of 13 cards each.
Each suit consists of one ace (A), nine numbered cards (2 to 10), one jack (J), one queen (Q) and one king (K). King, queen and Jack are face cards.
Four suits are spades, hearts, diamonds and clubs. Clubs and spades are of black colour while hearts and diamonds are red in colour.
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