Probability

Probability is a branch of mathematics that deals with the occurrence of a random event. For example, when a coin is tossed in the air, the possible outcomes are Head and Tail.

• Probability is a branch of mathematics that deals with calculating the likelihood of a given event's occurrence.
• Probability is the measure of the likeliness that an event will occur.
• Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).
• The higher the probability of an event, the more certain that the event will occur.

Importance of Probability

• There are branches of maths that equip you to make decisions in situations in which you have perfect information.
• Probability trains you to make decisions in situations which there are observable patterns, but a degree of uncertainty. Uncertainty and randomness occur in just about every field of application and in daily life for example probably the price of X share will go up, probably ‘X’ team will win the match , so it is extremely useful and interesting to understand probability.
• Probability is written as,
• What is the probability that an even number would come if we throw a dice?
Ans. Probability of the event would be -> number of ways it can be(3)/total number of outcome(6) = 0.50

Examples of Probability

Example 1: The chances of rolling a 4 with a die Number of ways it can happen : 1(There is only one face with a “4” on it) Total number of outcomes : There are 6 faces altogether
So, the probability = 1/6

Example 2: There are 5 marbles in a bag : 4 are blue and 1 is red. What is the probability that the blue marble gets picked?
Number of ways it can happen : 4(There are “4” blue) Total number of outcomes : There are 5 marbles in a bag
So, the probability = 4/5 = 0.8

Key Terms in Probability

• Statistic: A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic.
• Parameter: A parameter is a number that is a property of the population.
For example, Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.
• Experiment or Trial: An action where the result is uncertain.
For example: Tossing a coin, throwing dice, seeing what pizza people choose.
• Sample Space: The set of all possible outcomes of an experiment.
For Example: Selecting a card from a deck. There are 52 cards in a deck (excluding Jokers) Hence, the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }
• Sample Points: The Sample Space is made up of Sample Points. The elements of sample space are sample points.
For example: In the deck of cards- the 5 of Clubs is a sample point , the King of Hearts is a sample point "King" is not a sample point. As there are 4 Kings that is 4 different sample points.
• Event: A single result of an experiment
For Example :
1. Getting a Tail when tossing a coin is an event, getting a "5" when a die is rolled is an event.
2. Rolling an "even number" (2, 4 or 6) is also an event

Types of Events

There are two types of Events
1. Dependent Event
2. Independent Event

Dependent Event:

Dependent event also called "Conditional", where one event is affected by other events

Example: Drawing 2 Cards from a Deck. After taking one card from the deck there is one card less than the previous, so the probability changes!

Let's look at the chances of getting a King:
For the 1st card, the chance of drawing a King is 4 out of 52 But for the 2nd card, If the 1st card was a King, then the 2nd card is less likely to be a King, as only 3 of the 51 cards left are Kings. If the 1st card was not a King, then the 2nd card is slightly more likely to be a King, as 4 of the 51 cards left are King. This is because we are removing cards from the deck.

Independent Event:

One event is not affected by any other events.

Example:

You toss a coin three times and it comes up "Heads" each time. what is the chance that the next toss will also be a "Head"? The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin. What it did in the past will not affect the current toss.

Author:Mohit T (Algae Study)