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The sum of Probabilities is an important theorem used in the field of probability. Probability is a branch of mathematics that deals with the likeness or occurrence of an event
- The value is expressed from zero to one.
- To calculate the probability of a single event, we should first determine the number of samples.
- It is calculated by dividing the number of favourable outcomes by the total number of outcomes.
- The sum of Probabilities is the introductory probability proposition, which is also used in the probability distribution.
- The sum of the probability of the occurrence of an event and not the occurrence of an event is equal to 1.
- For example, the probability of a teacher coming for a lecture and not coming for the lecture is one.
- Mathematically, it can be represented as:
P(A) + P(A') = 1
- Where P(A): Probability of occurence of an event
- P(A’): Probability of not occurence of an event.
Key Terms: Sum of Probabilities, Probability, Sample Space, Events, Theoretical Probability, Experimental Probability, Complementary Events, Equally Likely Events, Axiomatic Probability
Sums of Probabilities of an Events
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Assume an event E can occur in r ways out of a sum of n probable or possible inversely likely ways. Also, the probability of passing of the event or its success is expressed as:
P (E) = r/ n
- The probability that the event won't do or known as its failure is expressed as:
P (E’) = (n-r)/ n = 1- (r/ n)
- E’ represents that the event won't happen.
- Thus, now we can say:
P (E) + P (E’) = 1
- This means that the aggregate of all the chances in any random test or trial is equal to 1.
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What is Probability?
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Probability is defined as a measure of the occurrence of an event. It is the ratio of favourable outcomes to a total number of outcomes. The probability of all the events in a sample space adds up to 1.
- Its value ranges from 0 to 1, where 0 means the event is an insolvable event and 1 indicates a certain event.
- Numerous events cannot be predicted with total certainty.
- We can prognosticate only the chance of an event, i.e., how likely they're to be, using it.
- The value of the events cannot be negative.
- It is used in the fields of engineering, statistics, financing, and other branches of mathematics.
Example of What is Probability?Example: For illustration, when we toss a coin, either we get Head OR Tail, only two possible issues are possible (H, T). But if we toss two coins in the air, there could be three possibilities of events to do, similar as both the coins show heads or both show tails or one shows heads and one tail, i.e. (H, H), (H, T), (T, T). |
Probability Formulas
Formula for Probability
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Probability of an event is equal to the rate of the number of favourable issues and the total number of issues. The formula for probability of an event is given as:
Probability of event to be P (E) = Number of favourable issues/ Total Number of issues
Example of Formula for ProbabilityExample: There are 8 pillows in a bed, 1 are red, 1 are yellow and 6 is blue. What is the probability of picking a yellow pillow? Ans: The probability is equal to the number of yellow pillows in the bed divided by the total number of pillows, i.e. 1/8 = 1/8. |
Probability Tree
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A probability tree is a picture indicating probabilities as well as the outcomes of an event. The tree is further divided into nodes and branches. Node is used for the representation of an event.
- Branches are used for the representation of the link between an event and its outcome.
- One of the easiest ways to break a probability problem is to construct a probability tree.
- Tree diagrams are a way of showing combinations of two or more events.
- Each branch is labelled at the end with its outgrowth, and the probability is written alongside the line.
- Two events are independent if the probability of the first event passing has no impact on the probability of the alternate event passing.
Probability Tree
Types of Probability
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The field of probability is divided into three categories which are as follows:
Theoretical Probability
Theoretical probability involves prediction about a particular event which can be precisely done with the access of statistical data of an event. Description of probability in statistics is grounded on the possibility of the circumstance of an outgrowth.
Example of Theoretical ProbabilityExample: Suppose if you're willing to find out the theoretical probability of getting a number'5'on rolling a die, also you should determine the number of possible issues.
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Experimental Probability
Experimental Probability is the description statistics of unlike theoretical probability description includes the number of trials.
Example of Experimental ProbabilityExample: Suppose a coin is tossed 30 times and out of those 30times we get tails 12 times, also the experimental probability of getting a head is 1230.
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Axiomatic Probability
Axiomatic Probability is a proposition of unifying probability where there's an operation of a set of rules made by Kolmogorov.
Types of Events
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The type of events are as follows:
Equally Likely Events
When the events have the same theoretical probability of happening, then they're called equally likely events. The results of a sample space are called equally likely if all of them have the same probability of being.
Example of Equally Likely EventsExample: For illustration, if you throw a die, then the probability of getting 1 is1/6. Also, the probability of getting all the figures from and 6, one at a time is1/6. Hence, the following are some illustrations of inversely likely events when throwing a die: Getting 3 and 5 on throwing a die. Getting an even number and an odd number on a die. Getting 1, 2 or 3 on rolling a die are equally likely events, since the chances of each event are equal. |
Complementary Events
Complementary Events are a type of event in which one outcome can only happen when the other does not. Each of these events complements the other.
Example of Complementary EventsExample 1: Like a person will come or not come to your house, getting a job, or not getting a job, etc. are cases of complementary events. Example 2: It'll rain or not rain today.
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| Class 10 Maths Related Concepts | ||
|---|---|---|
| Relation and Function | Analysis of Variance | Mode of Grouped Data |
| Types of Vectors | Types of events | Addition of Vectors |
Things to Remember
- Sum of the probabilities of all the outcomes must be equal to one.
- The probability tree will give the visual representation of the probability of an event.
- It is greater than zero and less than equal to one.
- Equally Likely events and complementary events are two types of events.
- Two events are said to be independent when the occurrence of an event is not dependent on another event.
Sample Questions
Ques: What is the likelihood of receiving a tail if a coin is tossed once? (3 marks)
Ans: The total number of possible outcomes is two, which are Head and Tail.
Let E represent the event of receiving a tail.
P(E)=No of outcomes favourable to E/Total no of outcomes of the experiment=12
Ques: A bag contains three identically sized and weighed blue, red, and yellow balls. What is the probability of getting a (I) blue ball, (II) yellow ball, and (III) red ball if Archana chooses a ball at random from the bag? (3 marks)
Ans: The bag contains three balls, one of which is red, one of which is blue, and one of which is yellow. If Archana draws a ball at random from the bag, then
(I) The chance of getting a blue ball is 1/3.
(II) The chance of getting a yellow ball is 1/3.
(III) The chance of getting a red ball is 1/3.
Ques: What is sum of probabilities? (2 marks)
Ans: It is proved that sum of probabilities is equal to 1. Probability of an event which does not occur is equal to 1 – Probability of the event which occurs. Probability that one or other event occurs is the sum of their individual probabilities.
Ques: What are the rules of Probability? (3 marks)
Ans: Rules of Probability are as follows:
- For an event A, 0 ≤ P(A) ≤ 1
- Sum of the probabilities of all possible outcomes is ONE.
- Complement Rule
- Probabilities involves Multiple Events
- Addition Rule
Ques: Can we add two probabilities? Give an example. (3 marks)
Ans: Yes, you can add two probabilities by multiplying them. Example:
Probability of first event is 2/7, and Probability of second event is 4/7.
Probability of both events is 2/7 * 4/7 = 8/49.
Ques: Two coins are flipped 40 times simultaneously. What is the probability of both coins landing on heads? (2 marks)
Ans: The possible outcomes - (H, H), (H, T), (T, H), and (T, T).
The no. of possible outcomes of both coins landing on heads is 1
So, the probability will be 1/4 or 25%.
Ques: What is the probability of not getting a 4 if you roll a dice? (3 marks)
Ans: There are 6 events that can occur. We can get 1,2,3,4,5,6 when we roll a dice.
- P(getting 4) = 1/6
- Since both the events are complimentary.
- Therefore P(getting 4) + P(not getting 4) = 1
- P(not getting 4) = 1-1/6
- P(not getting 4) = 5/6
Ques: Find the probability of getting a number less than 3 in a single throw of a die? (2 marks)
Ans: Possible outcome = {1, 2}
∴ P (Getting a number < 3) = 2/6 = 1/3
Ques: If P(E) = 0.06, what is the probability of 'not E'? (2 marks)
Ans: It is given that P(E) = 0.06
- P(E) + P (not E) = 1
- 0.06 + P (not E) = 1 ⇒ P (not E) = 1 – 0.06
- 0.94
Thus, probability of 'not E' = 0.94.
Ques: There are 20 balls in a bag out of which 10 are black, 2 are red, 5 is blue, 1 are pink, and 2 are purple. Let X be the event of selecting a primary color. Find P(X')? (3 marks)
Ans: X = {10 black, 5 blue}
- Total balls = 20
- Number of favorable outcomes = 15
- P(X) = 15/ 20
- Using the rule of complementary events, P(A') = 1 - P(A)
- P(X') = 1 - (15 / 20) = 5 / 20
Ques: Two dice are thrown simultaneously. What is the probability of obtaining a total of 5? (2 marks)
Ans: The possible outcomes are for a sum 5 are -
{ 1, 4 }, { 4, 1 }{ 3, 2 } and { 2, 3}
The total no. of outcomes are 36
So, the probability of obtaining a total of 5 is 4 / 36 = 1 / 9
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