The role of probability theory is to provide a framework for analyzing phenomena with uncertain outcomes. (Image by John Tsitsiklis.)
Author(s)
Prof. John Tsitsiklis
Prof. Patrick Jaillet
Probability definition is — the quality or state of being probable. How to use probability in a sentence. Windows 7 Ultimate Activator Rar Free Download. If you want to make your xp genuine type xp as command aswell as if you want vista type vista as command, and if.
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Instructor(s)
Prof.
As Taught In
Spring 2002
Course Number
2.24
Level
Undergraduate/Graduate
Features
Lecture Notes, Student Work
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This OCW supplemental resource provides material from outside the official MIT curriculum.
The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management. This resource is a companion site to 6.041SC Probabilistic Systems Analysis and Applied Probability. It covers the same content, using. Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. & #160;MLN 2018 is the first edition of the International Conference on Machine Learning for Networking. The goal of the conference is to provide a forum for scientists, engineers and researchers to discuss and exchange novel ideas, results, experiences and work-in-process on.
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The tools of probability theory, and of the related field of statistical inference, are the keys for being able to analyze and make sense of data. These tools underlie important advances in many fields, from the basic sciences to engineering and management.
This resource is a companion site to 6.041SC Probabilistic Systems Analysis and Applied Probability. It covers the same content, using videos developed for an edX version of the course.
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OCW Scholar Version
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More Lessons On Probability
Here we shall be looking into solving probability word problems involving:
- Probability and Sample Space
- Probability and Frequency Table
- Probability and Area
- Probability of Simple Events
- Probability and Permutations
- Probability and Combinations
- Probability of Independent Events
We will now look at some examples of probability problems.
Example:
At a car park there are 100 vehicles, 60 of which are cars, 30 are vans and the remainder are lorries. If every vehicle is equally likely to leave, find the probability of:
a) a van leaving first.
b) a lorry leaving first.
c) a car leaving second if either a lorry or van had left first.
Solution:
a) Let S be the sample space and A be the event of a van leaving first.
n(S) = 100
n(A) = 30
Probability of a van leaving first:
b) Let B be the event of a lorry leaving first.
n(B) = 100–60–30 = 10
Probability of a lorry leaving first:
c) If either a lorry or van had left first, then there would be 99 vehicles remaining, 60 of which are cars. Let T be the sample space and C be the event of a car leaving.
n(T) = 99
n(C) = 60
Probability of a car leaving after a lorry or van has left:
Example:
A survey was taken on 30 classes at a school to find the total number of left-handed students in each class. The table below shows the results:
No. of left-handed students 0 1 2 3 4 5 Frequency (no. of classes) 1 2 5 12 8 2
Probability 233 Definition
A class was selected at random.
a) Find the probability that the class has 2 left-handed students.
b) What is the probability that the class has at least 3 left-handed students?
c) Given that the total number of students in the 30 classes is 960, find the probability that a student randomly chosen from these 30 classes is left-handed.
Solution:
a) Let S be the sample space and A be the event of a class having 2 left-handed students.
n(S) = 30
n(A) = 5
Probability 233 Probability
b) Let B be the event of a class having at least 3 left-handed students.
n(B) = 12 + 8 + 2 = 22
c) First find the total number of left-handed students:
No. of left-handed students, x 0 1 2 3 4 5 Frequency, f (no. of classes) 1 2 5 12 8 2 fx 0 2 10 36 32 10
Total no. of left-handed students = 2 + 10 + 36 + 32 + 10 = 90
Here, the sample space is the total number of students in the 30 classes, which was given as 960.
Let T be the sample space and C be the event that a student is left-handed.
n(T) = 960
n(C) = 90
Probability And Area
Example:
ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN.
Solution:
Let 2x be the length of the square.
Area of square = 2x × 2x = 4x2
Area of triangle MCN is
This video shows some examples of probability based on area.
- Show Video Lesson
Probability 233 Calculator
Probability Of Simple Events
Probability 233 Math
The following video shows some examples of probability problems. A few examples of calculating the probability of simple events.
Example:
- What is the probability of the next person you meeting having a phone number that ends in 5?
- What is the probability of getting all heads if you flip 3 coins?
- What is the probability that the person you meet next has a birthday in February? (Non-leap year)
This video introduces probability and gives many examples to determine the probability of basic events.
Example:
- A bag contains 8 marbles numbered 1 to 8
a. What is the probability of selecting a 2 from the bag?
b. What is the probability of selecting an odd number?
c. What is the probability of selecting a number greater than 6? - Using a standard deck of cards, determine each probability.
a. P(face card)
b. P(5)
c. P(non face card)
Using Permutations To Solve Probability Problems
This video shows how to evaluate factorials, how to use permutations to solve probability problems, and how to determine the number of permutations with indistinguishable items.
A permutation is an arrangement or ordering. For a permutation, the order matters.
Examples:
- If a class has 28 students, how many different arrangements can 5 students give a presentation to the class?
- How many ways can the letters of the word PHEONIX be arranged?
- How many ways can you order 3 blue marbles, 4 red marbles and 5 green marbles? Marbles of the same color look identical.
- Show Video Lesson
Using Combinations To Solve Probability Problems
This video shows how to evaluate combinations and how to use combinations to solve probability problems.
A combination is a grouping or subset of items. For a combination, the order does not matter.
Examples:
- The soccer team has 20 players. There are always 11 players on the field. How many different groups of players can be in the field at the same time?
- A student needs 8 more classes to complete her degree. If she has met the prerequisites for all the courses, how many ways can she take 4 class next semester?
- There are 4 men and 5 women in a small office. The customer wants a site visit from a group of 2 men and 2 women. How many different groups van be formed from the office?
How To Find The Probability Of Different Events?
This video explains how to determine the probability of different events. This can be found that can be found using combinations and basic probability.
- The probability of drawing 2 cards that are both face cards.
- The probability of drawing 2 cards that are both aces.
- The probability of drawing 4 cards all from the same suite.
A group of 10 students made up of 6 females and 4 males form a committee of 4.
What is the probability the committee is all male?
What is the probability that the committee is all female?
What is the probability the committee is made up of 2 females and 2 males?
- Show Video Lesson
How To Find The Probability Of Multiple Independent Events?
This video explains the counting principle and how to determine the number of ways multiple independent events can occur.
Examples:
- How many ways can students answer a 3-question true of false quiz?
- How many passwords using 6 digits where the first digit must be letters and the last four digits must be numbers?
- A restaurant offers a dinner special in which you get to pick 1 item from 4 different categories. How many different meals are possible?
- A door lock on a classroom requires entry of 4 digits. All digits must be numbers, but the digits can not be repeated. How many unique codes are possible?
How To Find The Probability Of A Union Of Two Events?
This video shows how to determine the probability of a union of two events.
Examples:
- If you roll 2 dice at the same time, what is the probability the sum is 6 or a pair of odd numbers?
- What is the probability of selecting 1 card that is red or a face card?
- Show Video Lesson
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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