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probability and chance

Published on Nov 26, 2015

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probability and chance

AN EXPLORATION IN RANDOMNESS
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probability

The likelihood of a specific event occurring.
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experiment

Describes a situation involving random outcome(s)
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Examples of experiments

  • Rolling a pair of dice
  • Drawing a card from a deck at random
  • Measuring heights of 100 people chosen randomly
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Random

Equally likely occurrence of all possible events without choice
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Outcome

A specific result of an experiment
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Example outcomes

  • Getting doubles on a pair of dice
  • Drawing an Ace of Spades at random
  • Measuring a random person's height at 5'5"
Photo by Justin.Taylor

Sample space

The collection of all possible outcomes of an experiment

example

  • Experiment:  flipping two coins
  • Sample space:  S = {HH, HT, TH, TT}
  • There are four outcomes in this sample space.

Event

A specific outcome of interest in an experiment

EXAMPLE: Wanting to roll a YAHTZEE! when rolling a set of 5 dice at random (experiment)

Properties of probability

  • The probability of event A occurring is written as P(A).
  • For any event A, P(A) must always between 0 and 1. 
  • P(impossible event) = 0, and P(certain event) = 1. 
  • P(not A) = 1 - P(A).  
  • P(not A) is known as the complement of A.  

P(A) is the ratio of all favorable outcomes (ways event can happen) over all possible events in sample space.

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example

  • Desired event:  rolling an even number on a  die
  • # of favorable outcomes:  3
  • Total possible outcomes:  6
  • P(even #) = 3/6 or 1/2 (0.5, or 50%)
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LAW OF TOTAL PROBABILITY: The sum of all probabilities of events in a sample space must always add to 1.

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compound probability

Probability of multiple events occurring in succession

Tree Diagram

Visual way to interpret compound probabilities

independent events

Outcomes do not affect one another
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examples

  • Rolling a pair of dice
  • Flipping a coin multiple times
  • Drawing numbers from a hat WITH replacement
Photo by Micah Sittig

dependent events

One outcome affects the other

examples

  • Drawing three of the same card from a deck
  • Pulling two of the same letter in Scrabble
  • Getting an ace and a face card in Blackjack

random variable

Value is the numerical outcome of an event
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Example

  • Experiment:  rolling a die 
  • Random variable X = number rolled on die
  • Sample space S = {1, 2, 3, 4, 5, 6} 
  • These are values X can take on as a RV 
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PROBABILITY DISTRIBUTION: Represents all outcomes of experiment as a function of the random variable

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relative frequency diagram

Graph representing probability distribution

Theoretical proability

Probability we expect based on the math
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Experimental probability

Probability observed after multiple trials of experiment
Photo by Serge Melki