Probability Rules

Probability Rules

You must have played cards and enjoyed that too.

Girl won the game of cards with aces

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


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?

1. Addition Rule

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).

\(\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B} \mid \mathrm{A})\)

 
Thinking out of the box
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? 

Combinations for ice cream flavours and cones

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 \( 3 \times 6 = 18 \)

 
important notes to remember
Important Notes
  1. Probability of occurrence of an event P(E) = Number of favorable outcomes/Total Number of outcomes.

  2. The maximum probability of an event is its sample space. The probability of any event exists between \(0\) and \(1\).

  3. In independent events, the outcome of one event doesn't affect the outcome of other events, whereas in dependent events, the outcome of one event influences the outcome of another event.

Solved Examples

Example 1

 

 

Ben is fond of reading books.

He often goes to the library.

Ben checking for books in library

The probability that he checks out:

(a) a work of fiction is \(0.40\),

(b) a work of non-fiction is \(0.30\),  

(c) both fiction and non-fiction is \(0.20\)

What is the probability that he checks out both fiction as well as non fiction?

Solution

Let F be the event for Ben checking out fiction.

Let N be the event for Ben checking out non-fiction.

Then, based on the rule of addition:

\(P(F \cup N)=P(F)+P(N)-P(F \cap N)\)
\(\mathrm{P}(\mathrm{F} \cup \mathrm{N})=0.40+0.30-0.20=0.50\)

\(\therefore\) Probability that he checks out both is \(0.5\)
Example 2

 

 

A jar contains 4 green marbles and 6 yellow marbles.

Two marbles have been drawn from the jar.

The second marble has been drawn without replacement.

What is the probability that both the drawn marbles will be yellow?

6 yellow and 4 green marbles

Solution

Let A = the event that the first marble is yellow; and let B = the event that the second marble is yellow. We know the following:

  • In the beginning, there are 10 marbles in the box, 6 of which are yellow. Therefore, P(A) = 6/10
  • After the first selection, there are 9 marbles in the jar, 5 of which are yellow. Therefore, P(B|A) = 5/9

Therefore, based on the rule of multiplication:

\(P(A \cap B)=P(A). P(B \mid A)\)
\(\mathrm{P}(\mathrm{A} \cap \mathrm{B})=(6 / 10)^{*}(5 / 9)=30 / 90=1 / 3=0.33\)

\(\therefore\) Probability that both marbles are yellow = \(0.33\)
Example 3

 

 

Out of a deck of \(52\) cards, Kate has to draw two cards consecutively, without replacement.

Deck of cards

She asked Jane to calculate the probability of drawing a king and a queen consecutively.

Let's help Jane to calculate the probability.

Solution

Total number of events = total number of cards = \(52\)

Probability of drawing a queen = 4/52 = 1/13

Now, the total number of cards = \(51\)

Probability of drawing a king = 4/51

So, the probability of drawing a king and a queen consecutively, without replacement = 1/13 * 4/51 = 4/ 663

\(\therefore\) Probability is 4/663
Example 4

 

 

There are \(6\) children in a classroom and \(6\) benches for them to sit.

Their teacher makes them sit at a different place every month. In how many ways can she make them sit in the classroom?

Combinations of seating arrangement

Solution

There are \(6\) children and \(6\) benches for them to sit. 

Hence, their teacher will apply the fundamental counting principle to find the number of ways in which she can make them sit. 

The number of ways in which she can make the children sit in the classroom is: \( 6 \times 6 = 36 \)

\(\therefore\) There are \(36\) ways.
Example 5

 

 

While going through the class records, the teacher got the following information:

  • 40 % of the students study math and science.
  • 60 % of the students study science.

What is the probability of students studying math, given that the student is already studying science?

Solution

Probability of students studying math and science = P(M&S) = \(0.40\)

Probability of students studying math = P(S) = \(0.60\)

Probability of students studying math, given that he/she is already studying science = \(\mathrm{P}(\mathrm{S} \mid \mathrm{M}) / \mathrm{P}(\mathrm{S})\)

 =\(0.40/0.60\) =\( 2/3\) = \(0.67\)

\(\therefore\) Probability is \(0.67\)

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 
 
 

 

 


Let's Summarize

The mini-lesson targeted the fascinating concept of Probability Rules. The math journey around Probability Rules starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. What is the and/or rule in probability?

When events are mutually exclusive and we want to know the probability of getting one event OR another, then we can use the OR rule, that is, P(A or B) = P(A) + P(B). When events are independent and we want to know the probability of both the events occurring simultaneously, then we can use the AND rule, P(A and B)=P(A)⋅P(B).

2. What is the probability formula?

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

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