A branch of mathematics that deals with the happening of a random event is termed probability. It is used in Maths to predict how likely events are to happen.
The probability of any event can only be between 0 and 1 and it can also be written in the form of a percentage.
The probability of event A is generally written as P(A).
Here P represents the possibility and A represents the event. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty
If we are not sure about the outcome of an event, we take help of the probabilities of certain outcomes—how likely they occur. For a proper understanding of probability we take an example as tossing a coin:
There will be two possible outcomes—heads or tails.
The probability of getting heads is half. You might already know that the probability is half/half or 50% as the event is an equally likely event and is complementary so the possibility of getting heads or tails is 50%.
Formula of Probability
Probability of an event, P(A) = Favorable outcomes / Total number of outcomes
Some Terms of Probability Theory
- Experiment: An operation or trial done to produce an outcome is called an experiment.
- Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
- Favourable Outcome: An event that has produced the required result is called a favourable outcome. For example, If we roll two dice at the same time then the possible or favourable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
- Trial: A trial means doing a random experiment.
- Random Experiment: A random experiment is an experiment that has a well-defined set of outcomes. For example, when we toss a coin, we would get ahead or tail but we are not sure about the outcome that which one will appear.
- Event: An event is the outcome of a random experiment.
- Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when we toss a coin, there are equal chances of getting a head or a tail.
- Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
- Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. We cannot experience the same weather again and again.
- Complementary Events: The Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.
Some Probability Formulas
Addition rule: Union of two events, say A and B, then
P(A or B) = P(A) + P(B) – P(A∩B)
P(A ∪ B) = P(A) + P(B) – P(A∩B)
Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example – if A and B are two possible events, then
P(B) = 1 – P(A) or P(A’) = 1 – P(A).
P(A) + P(A′) = 1.
Conditional rule: When the probability of an event is given and the second is required for which first is given, then
P(B, given A) = P(A and B), P(A, given B). It can be vice versa
P(B∣A) = P(A∩B)/P(A)
Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. Then
P(A and B) = P(A)⋅P(B).
P(A∩B) = P(A)⋅P(B∣A)
Find the probability of getting a number less than 5 in a single dice throw.
Solution :
When the dice is rolled then there will be 6 outcomes.
Total number of favorable outcome { set of outcome } = {1, 2, 3, 4, 5, 6 }
= 6
Now as per the question
Probability of getting a number less than 5 in a single throw is 4
Numbers less than 5 are { 1,2,3,4}
therefore favorable outcome will be = 4
P(A) = Favorable outcomes / Total number of outcomes
= 4/6
= 2/3
Hence the probability of getting a number less than 5 in a single throw of a die is 2/3
Similar Questions
Question 1: Find the probability of getting a number less than 4 in a single throw of a die.
Solution:
When the dice is rolled then there will be 6 outcomes
Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 }
= 6
Now as per the question
probability of getting a number less than 4 in a single throw is 3
Numbers less than 4 are { 1,2,3}
Therefore favorable outcome will be = 3
P(A) = Favorable outcomes / Total number of outcomes
= 3/6
= 1/2
Hence the probability of getting a number less than 4 in a single throw of a die is 1/2
Question 2: Find the probability of getting a number more than 4 in a single throw of a die.
Solution:
When the dice is rolled then there will be 6 outcomes.
Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 }
= 6
Now as per the question
probability of getting a number more than 4 in a single throw is 2
Numbers more than 4 are { 5,6}
Therefore favorable outcome will be = 2
P(A) = Favorable outcomes / Total number of outcomes
= 2/6
= 1/3
Hence the probability of getting a number more than 4 in a single throw of a die is 1/3
Question 3: Find the probability of getting a number 5 in a single throw of a die.
Solution:
When the dice is rolled then there will be 6 outcomes.
Total number of favorable outcome { set of outcome } = {1 ,2 ,3 ,4 , 5, 6 }
= 6
Now as per the question
probability of getting a number 5 in a single throw is 1
Therefore favorable outcome will be = 1
P(A) = Favorable outcomes / Total number of outcomes
= 1/6
Hence the probability of getting a number 5 in a single throw of a die is 1/6
Question 4: What is the chance of rolling a 3 two times in a row?
Solution:
P(A) = Favorable outcomes / Total number of outcomes
Probability of getting 3 = 1/6.
Rolling dice is an independent event, it is not dependent on how many times it’s been rolled.
Probability of getting 3 two times in a row = probability of getting 3 first time × probability of getting 3 second time.
Probability of getting 3 two times in a row = (1/6) × (1/6) = 1/36.
Hence, the probability of getting 3 two times in a row 2.77 %.
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In daily life, usually the word ‘probably’ is used when people are not sure about certain things. For example, Probably, India may win the match today. There may be a chance that India may win or lose or maybe the match is a tie. This type of statement leads to the uncertainty of the event. The word Probability is formed from the word word ‘probably’, which means that when people are not sure about an event is happens or not. People have a proper method to find out the probability which will be discussed in this article.
Terms used in Probability
- Random Experiment: In the random experiment, we can not predict the result in advance. For example, if we toss a coin we can not predict that the head will appear, the tail also may appear.
- Event: Collection of some outcome of an experiment is known as an event.
- Sample Space: It is the collection of all possible outcomes. Suppose a dice is rolled out then the possible outcome is 1, 2, 3, 4, 5, or 6. It is denoted by S. S = (1, 2, 3, 4, 5, 6)
- Rolling of Dice: A dice is a solid cube shape. It has 6 square faces. The six faces are marked by 1, 2, 3, 4, 5, 6 dots. When a fair dice is rolled, then the total possible outcome is 1, 2, 3, 4, 5, or 6. So all these numbers are known as sample space.
Probability
The probability of an event is defined as the ratio of favorable outcomes to the sample space or total outcomes. We represent it by ‘P’.
Probability of an event (P) = ( Number of Favourable outcomes) / (Total number possible outcomes)
Solution:
Concept: To solve the given problem, follow the steps given below.
Step 1: First of all find out all possible outcomes of the given event. Represent it by S.
Step 2: Specify the number of favorable outcomes.
Step 3: Use the formula, Probability of an event = (Favorable outcomes) / (Total number of possible outcomes)
Step 4: Simplify and get the final answer.
When a dice is rolled, all possible outcomes are 1, 2, 3, 4, 5, 6.
We call it as sample space, S = (1, 2, 3, 4, 5, 6)
So total number of possible outcomes = 6
Favorable outcome (Required outcome) = 1
(Only 1 is smaller than 2, remaining number is greater than 2 so we will not consider them as favorable outcomes.)
So total number of favorable outcomes = 1
Probability = (Total number of favorable outcomes)/(Total number of possible outcomes)
Probability = 1/6
So, the probability of the given statement is 1/6.
Similar Questions
Question 1: What is the probability of rolling a number greater than 4 on a dice?
Solution:
When a dice is rolled, all possible outcomes are 1, 2, 3, 4, 5, 6.
S = (1, 2, 3, 4, 5, 6)
Number of possible outcomes, n(S) = 6
Favorable outcomes = (5, 6)
(Only 5 and 6 is greater than 4, so these two will be favorable cases)
Number of favorable outcomes, n(F) = 2
Probability = (Number of favorable outcomes)/(Number of total outcomes)
Probability = 2/6
=1/3
So, the probability of the given statement is 1/3.
Question 2: What is the probability of rolling an odd number on a dice?
Solution:
When a dice is rolled, all possible outcomes are 1, 2, 3, 4, 5, 6.
S = (1, 2, 3, 4, 5, 6)
Number of possible outcomes, n(S) = 6
Favorable outcomes = (1, 3, 5)
(Only 1, 3, 5 are the odd number obtained when a dice is rolled)
Total number of favorable outcomes, n(F) = 3
Probability = (Number of favorable outcomes)/(Number of total outcomes)
Probability = 3/6
= 1/2
So, the required probability is 1/2.
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