# Probability Rules

Probability Rules

You must have played cards and enjoyed that too. While playing a game of cards, did you ever predict the next card to be a king, a queen or an ace, which could make you win the game?

Or may be you wished for a certain card to come up next which could be in your favor!

Have you ever thought that you have been using the concept of probability in your daily life, whether it is the game of cards or anything else?

There are conditions or possibilities of events occurring at the same time or you chance upon events going consecutively, so how do you find out the probability of those events?

Well! There are probability rules that you can follow!

Let's explore the simulation below to get an idea about probability! Enter the values to calculate the probability of numbers.

This mini lesson will tell you about probability rules, the complement rule and the fundamental counting principle.

Check out the interesting examples and a few interactive questions at the end of the page.

## Lesson Plan

 1 What Is Probability? 2 Thinking Out of the Box! 3 Important Notes on Probability Rules 4 Solved Examples on Probability Rules 5 Interactive Questions on Probability Rules

## What Is Probability?

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty.

Using probability, one can predict only the chance of an event to occur, i.e., how likely they are going to happen.

For example, when a coin is tossed, there is a probability to get heads or tails.

Properties:

• Probability of an impossible event is phi or a null set.
• The maximum probability of an event is its sample space (sample space is the total number of possible outcomes)
• Probability of any event exists between 0 and 1. (0 can also be a probability).
• There cannot be a negative probability for an event.
• If A and B are two mutually exclusive outcomes (Two events that cannot occur at the same time), then the probability of A or B occurring is the probability of A plus the probability of B.

The probability formula is the ratio of the possibility of occurrence of an outcome to the total number of outcomes.

Probability of occurrence of an event P(E) = Number of favorable outcomes/Total Number of outcomes.

## What Are the Rules of Probability in Math?

Whenever an event is the union of two other events, say A and B, then $$P(A \text { or } B)=P(A)+P(B)-P(A \cap B)$$

$$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$$

2. Complementary Rule

Whenever an event is the complement of another event, specifically, if A is an event, then P(not A)=1−P(A) or P(A') = 1 - P(A').

$$P(A)+P\left(A^{\prime}\right)=1$$

3. Conditional Rule

When event A is already known to have occurred and probability of event B is desired, then P(B, given A)=P(A and B)P(A, given B). It can be vica versa in case of event B.
$$\mathrm{P}(\mathrm{B} \mid \mathrm{A})=\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \mathrm{P}(\mathrm{A})$$

4. Multiplication Rule

Whenever an event is the intersection of two other events, that is, events A and B need to occur simultaneously. Then P(A and B)=P(A)⋅P(B).

$$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B} \mid \mathrm{A})$$ Think Tank
1. A random number is chosen from $$1$$ to $$100$$. What is the probability that it is a multiple of $$11$$?
2. Tim rolls a fair die. What is the probability of die, not landing on $$4$$?

## What Is the Fundamental Counting Principle?

The fundamental counting principle is a rule which counts all the possible ways for an event to happen or the total number of possible outcomes in a situation.

It states that when there are $$n$$ ways to do one thing, and $$m$$ ways to do another thing, then the number of ways to do both the things can be obtained by taking their product. This is expressed as $$n \times m$$.

Example:

An ice cream seller sells 3 flavors of ice creams, vanilla, chocolate and strawberry giving his customers 6 different choices of cones.

How many choices of ice creams does Wendy have if she goes to this ice cream seller? Solution

Wendy has 3 choices for the ice cream flavors and 6 choices for ice cream cones.

Hence, by the fundamental counting principle, the number of choices that Wendy has can be represented as $$3 \times 6 = 18$$

More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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