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Probability · Lecture 3 of 76 · 8:42

Geniuses and Chocolates

Geniuses and Chocolates on YouTube

Study guide

What this lecture covers

This recitation problem practices the properties of probability laws that follow from the axioms, such as P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Given that 60% of a class are geniuses, 70% love chocolate, and 40% are both, the task is to find the probability that a randomly chosen student is neither.

Instead of plugging into the union formula directly, the video works the problem visually, using a Venn diagram cut into a partition of disjoint regions and building up from there.

Key ideas

  • Venn diagrams: a useful tool for visualizing how overlapping events and their probabilities relate to each other.
  • Partitions: cutting the sample space into disjoint pieces that together cover everything, which makes it possible to add their probabilities.
  • Reading off regions: the diagram is split into four labeled regions — neither event, only "genius," only "chocolate lover," and both.
  • Working from known unions: P(genius) = P(only genius) + P(both) and similarly for chocolate lovers, letting you solve for the unknown regions by simple algebra.
  • Using the normalization axiom: since all four regions must sum to 1, the last unknown region (neither event) is found by subtraction.

Before you watch

  • Know the axioms of probability and the idea of disjoint events, from the first lecture in this course.
  • Some familiarity with Venn diagrams is helpful but not required.

Check your understanding

  1. Why must the four regions in the Venn diagram (neither, only genius, only chocolate lover, both) be disjoint before you can add their probabilities?
  2. How is P(genius) expressed in terms of the "only genius" and "both" regions, and why?
  3. Once you know the probabilities of three of the four regions, how do you find the fourth?
  4. How would the answer change if geniuses and chocolate lovers were completely disjoint (0% overlap)?

Vocabulary

Venn diagram (noun)
A picture using overlapping circles to show how sets or events relate to each other.
A Venn diagram makes the overlap between two events easy to see.
overlap (noun)
The part that two things share in common.
40% of students are in the overlap between geniuses and chocolate lovers.
region (noun)
A distinct labeled part of a diagram or sample space.
The Venn diagram is split into four separate regions.
algebra (noun)
The use of symbols and equations to solve for unknown values.
Simple algebra finds the size of each unknown region.
subtraction (noun)
The mathematical operation of taking one quantity away from another.
The last region is found by subtraction from 1.
genius (noun)
Here, a label for a student in one of the two overlapping categories in the problem.
60% of the class are labeled geniuses in this example.
randomly chosen (phrase)
Picked without any pattern, so every option has a fair chance.
The question asks about a randomly chosen student.
neither (adjective)
Not one and not the other, of two options.
The neither region contains students who like neither thing.
unknown (adjective)
Not yet known, and needing to be found by calculation.
One unknown region remains after three are found.
property (noun)
A characteristic or rule that something follows.
The properties of probability laws let you find P(A union B).
practice (a skill) (verb)
To repeatedly work on a skill to become better at it.
This problem is used to practice probability properties.
visually (adverb)
By using a picture rather than only numbers or symbols.
The problem is worked visually using a Venn diagram.
cut into (phrasal verb)
To divide something into separate pieces.
The diagram is cut into a partition of disjoint regions.
build up from (phrasal verb)
To start from a simple base and add more steps.
The solution builds up from the smallest known region.
read off (phrasal verb)
To find a value directly by looking at a diagram or table.
You can read off each region's probability from the diagram.
disjoint (adjective)
Having no outcomes in common.
The four labeled regions are disjoint.
overlap probability (noun)
The chance that two events both happen at once.
The overlap probability of 40% is given directly in the problem.
solve for (phrasal verb)
To find the value of an unknown quantity in an equation.
You solve for the last region using the normalization axiom.
familiarity (noun)
Knowledge or comfort gained through previous exposure to something.
Some familiarity with Venn diagrams helps but isn't required.
task (noun)
A specific piece of work that needs to be completed.
The task is to find the probability of neither event.

Chapters

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Katie Szeto

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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