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Probability

Probability, as the name suggests, deals with uncertainty. Probability of an event is defined as a positive number which denotes the chances of happening of an event. If an event is certain, i.e. there are 100% chances of that event to occur, then the probability of such event will be one. On the other hand, for an impossible event, i.e. there are no chances of that event to occur, the probability will be zero. Thus, we may summarize as:

1) Probability of occurrence of an event is a number lying between 0 and 1.

2) Probability of a certain event = 1.

3)
Probability of an impossible event = 0

4)
The sum of happening probability of an event and not happening of an event is equal to $1$.

There are several terms and concepts related to this vast topic. In this page, we shall go ahead and learn about the probability and its applications in detail.

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Probability Help Topics

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Tutorvista helps you to understand each and every concept with details. Below is the list of main topics covered by our probability help program:

  • Complimentary events
  • Conditional probability
  • Compound events
  • Independent and Dependent events
  • Mutually exclusive events

Besides these main topics, there is another range of probability topics included in our help program. Understand all these concepts with the best probability tutor. Also Tutorvista provides tutoring for all grades.

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Definition

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Probability in mathematics is the calculation of uncertainty. Probabilities are not at a single point. It is a number expressing the chances for a specific event will occur. Probability table is used to tabulate all the data calculated for a standard values from that values new probabilities can be determined. The events can also be tabulated. Learn more on facts of the concept with Tutorvista now.

Terms

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Basic terms of probability are as follows:

Probability: Probability is the measure of how likely the events are occur.

1) Experiment: Experiment is a process by which an outcome is obtained

For Example: Tossing a coin

2) Sample Space: Sample space is the set $S$ of all possible outcomes of an experiment.

For Example: The sample space for a coin toss is, $S$ = $ {H,\ T}$

3) Trial: Each observation of an experiment is a trial.

4) Event: Event is any subset of the sample space.

5) Dependent Events: Occurrence of one event does have an effect on the probability that a second event will occur.

6) Independent Events: Occurrence of one event has no effect on the probability that a second event will occur

7) Outcome: Each result of a trial is called outcome of that trial.

Formulas

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Some basic probability formulas are given below:

1) Probability Formula:

$\Rightarrow\ P(A)$ = $\frac{\text{Number of outcomes favorable}}{\text{Total numbers of outcomes}}$

2) Addition Rule:

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

for P(A $\cap$ B) $\neq$ 0

3) Multiplicative Rule:

For independent events

P(A $\cap$ B) = P(A) $\times$ P(B)

For dependent events

P(A $\cap$ B) = P(A) $\times$ P(B|A)

4) Bayes Theorem


$P\left ( \frac{E_{i}}{A} \right )=\frac{P(E_{i})P\left ( \frac{A}{E_{i}} \right )}{\sum_{i=1}^{n}P(E_{i})P\left ( \frac{A}{E_{i}} \right )}$

where $E_i$ are n-mutually exclusive events with $P(E_i)$ ≠ 0

5) Binomial Distribution:

$P(x$ = $k)$ = $C(n,\ k)p^k\ (1\ -\ p)^{n\ -\ k}$

where $C(n,\ k)$ = $\frac{n!}{k!(n - k)!}$

Rules

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Different rules are followed to find the probability of an event. These are also called as law of probability.

Rule 1: Rule of Subtraction

The probability that event A will occur is equal to 1 minus the probability that event A will not occur.
$\Rightarrow$ P(A) = 1 - P(A')

Rule 2: Rule of Multiplication
The probability the events A and B both occur is equal to the probability that event A occurs times the probability that Event B occurs, given that A has occurred.
$\Rightarrow$ P(A $\cap$ B) = P(A) P(B|A)

Rule 3: Rule of Addition
The probability the event A or event B occurs is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.
$\Rightarrow$ P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)

Probability Simulations

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The probability simulations learning utilize the concepts of theoretical and experimental probabilities to solve problems involving uncertainty. The probability simulations learning to perform the model of real-world situations involving uncertainty. We can conduct probability simulations of the results for many problems of objects we pick should have the same number of results as the number of possible results of the problem, and all outcomes should be equally likely.

Theoretical and Experimental Probability


Theoretical probabilities of operations are performed in mathematically and describe what should happen. Experimental probability is calculated using data from tests or experiments. Experimental probability is the ratio of the number of times an outcome happened to the total number of events or trials. This type of ratio is labeled as the relative frequency.

If all the outcomes in a sample space S are equally likely, and E is an event within that sample space, then the theoretical probability of the event E is given by

Theoretical Probability Formula 

P(Event) = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Experimental probability = $\frac{(Frequency\ of\ an\ outcome)}{(Total\ number\ of\ trials)}$

Types

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There are four types of probability are marginal probability, conditional probability, joint probability and union probability. The general law of multiplication used to compute joint probability and The general law of addition is used to compute the probability of union. The conditional law is used to compute conditional probability.
A probability tree is a diagram indicating probabilities and some conditional probabilities for combination of two or more events. Probability tree is one of the simple and easiest ways to solve a probability problems.

Venn Diagram

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Venn diagrams are diagrams that show all possible logical relations between a finite collection. To draw a Venn diagram, first we draw a rectangle which is called your "universe". A Venn diagram is a picture that represents the outcomes of an experiment. The rectangle represents the sample space, which is all of the possible outcomes of the experiment. The probability the event A or event B occurs is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that both events A and B occur.

$\Rightarrow$ P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)

This can be shown on a Venn diagram. The rectangle represents the sample space or universe, which is all of the possible outcomes of the experiment. The circle labelled as A represents event A. Similarly for B.

Venn Diagram

So, in the diagram, P(A) + P(B) is the whole of A (the whole circle) + the whole of B (so we have counted the middle bit twice). A' is the complement of A and means everything not in A. So P(A') is the probability that A does not occur.

Examples

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Given below are some of the word problems on probability.

Solved Examples

Question 1: A bag contains 10 identical balls, of which 4 are blue and 6 are red. Three balls are taken out at random from the bag one after the other. Find the probability that all the three balls drawn are red.
Solution:
Total number of balls = 10

Step 1:
Since six out of ten balls are red

=> Probability that the first ball drawn is red = $\frac{ 6}{10}$.

Step 2:
If the first ball drawn is red, 9 balls are left and 5 out of them are red.

Similarly if the first two balls drawn are red colored, then the probability that the third ball drawn is red is $\frac{4}{8}$.

Hence by multiplication theorem, the required probability is

= $\frac{ 6}{10}$ * $\frac{4}{8}$

= $\frac{1}{6}$.

Question 2: James and John were selected for the final round of selection process to appoint a Vice President of the organization. The probability of John getting appointed as the Vice President is 0.47. What is the probability of James getting selected for the post?
Solution:
The event of James getting selected for the post is same as the event of John not getting appointed to the post. John not getting appointed to the post is the complementary event of John getting appointed.

The probability of John getting appointed as the Vice President = 0.47.

P(John not appointed) = 1 - 0.47 = 0.53.

Probability of James getting selected = 0.53.

Practice Problems

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Given below are some of the practice problems on Probability.

Practice Problems

Question 1: In a urn there is 4 red, 2 blue balls. If four balls are drawn from the urn contain 14 balls at random, what is the probability that 2 are red and 2 are blue.
Question 2: A box contain 8 red, 8 blue and 6 green balls. One ball is picked up randomly, what is the probability that it is neither red nor green.
More topics in Probability
Probability Homework Help Poisson Distribution
Sample Space Tree Diagram (Probability Theory)
Compound Probability Permutations and Combinations
Complementary Events Conditional Probability
Dependent Events Independent Events
Mutually Exclusive Events Theoretical Probability
Probability Problems Binomial Probability
Random Variables Simple Event
Compound Events Properties of Probability
Bayesian Probability Probability Axioms
Boole's Inequality Disjoint Events
Contingency Table Union of Events
Intersection of Events Probability Line
Types of Events Theoretical and Experimental Probability
Probability without Replacement Convergence in Probability
Multiplication Law of Probability Probability Density Function
Elementary Probability Coin Toss Probability
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