Probability. ◮ The probability of getting a straight is around 1 in 250. Straight: Consecutive cards, suit doesn’t matter, e.g., ◮ The probability of rolling snake eyes is 1/36. How many snake eyes? one pip and one pip. 1 ∗ 1 = 1. ◮ The probability that a poll of a 1000 people will report at least 50% support for a candidate with 60% support is 80%.
Terminology ◮ “Heads or Tails” in coin tossing (or coin flipping).
Terminology ◮ “Heads or Tails” in coin tossing (or coin flipping). ◮ One side of a coin is called the “heads” side and the other the “tails” side.
Terminology ◮ “Heads or Tails” in coin tossing (or coin flipping). ◮ One side of a coin is called the “heads” side and the other the “tails” side. ◮ Thus, one says that one gets one heads or one tails in a coin flip,
Terminology ◮ “Heads or Tails” in coin tossing (or coin flipping). ◮ One side of a coin is called the “heads” side and the other the “tails” side. ◮ Thus, one says that one gets one heads or one tails in a coin flip, or that the coin flip is heads or tails. ◮ Rolling one die or two dice.
Terminology ◮ “Heads or Tails” in coin tossing (or coin flipping). ◮ One side of a coin is called the “heads” side and the other the “tails” side. ◮ Thus, one says that one gets one heads or one tails in a coin flip, or that the coin flip is heads or tails. ◮ Rolling one die or two dice. ◮ Singular is die and plural is dice.
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...1 / 2.
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...1 / 2. ◮ The probability that the next person through the door is younger than 21 is 80%.
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...1 / 2. ◮ The probability that the next person through the door is younger than 21 is 80%. ◮ Amanda Knox is innocent with 70% probability. They are statements about a probability space.
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...1 / 2. ◮ The probability that the next person through the door is younger than 21 is 80%. ◮ Amanda Knox is innocent with 70% probability. They are statements about a probability space. (Except perhaps the last one or two.)
Probability ◮ If you flip a fair coin and get 50 heads, you will get heads the next time with probability ...1 / 2. ◮ The probability that the next person through the door is younger than 21 is 80%. ◮ Amanda Knox is innocent with 70% probability. They are statements about a probability space. (Except perhaps the last one or two.) Random experiment constructed by us, or the world.
Probability Space. 1. A “random experiment”:
Probability Space. 1. A “random experiment”: (a) Flip a biased coin;
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins;
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand.
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω .
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ;
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ;
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | =
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4;
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4; (c) Ω = { A ♠ A ♦ A ♣ A ♥ K ♠ , A ♠ A ♦ A ♣ A ♥ Q ♠ ,... } | Ω | =
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4; (c) Ω = { A ♠ A ♦ A ♣ A ♥ K ♠ , A ♠ A ♦ A ♣ A ♥ Q ♠ ,... } � 52 � | Ω | = . 5
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4; (c) Ω = { A ♠ A ♦ A ♣ A ♥ K ♠ , A ♠ A ♦ A ♣ A ♥ Q ♠ ,... } � 52 � | Ω | = . 5 3. Assign a probability to each outcome: Pr : Ω → [ 0 , 1 ] . (a) Pr [ H ] = p , Pr [ T ] = 1 − p for some p ∈ [ 0 , 1 ]
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4; (c) Ω = { A ♠ A ♦ A ♣ A ♥ K ♠ , A ♠ A ♦ A ♣ A ♥ Q ♠ ,... } � 52 � | Ω | = . 5 3. Assign a probability to each outcome: Pr : Ω → [ 0 , 1 ] . (a) Pr [ H ] = p , Pr [ T ] = 1 − p for some p ∈ [ 0 , 1 ] (b) Pr [ HH ] = Pr [ HT ] = Pr [ TH ] = Pr [ TT ] = 1 4
Probability Space. 1. A “random experiment”: (a) Flip a biased coin; (b) Flip two fair coins; (c) Deal a poker hand. 2. A set of possible outcomes: Ω . (a) Ω = { H , T } ; (b) Ω = { HH , HT , TH , TT } ; | Ω | = 4; (c) Ω = { A ♠ A ♦ A ♣ A ♥ K ♠ , A ♠ A ♦ A ♣ A ♥ Q ♠ ,... } � 52 � | Ω | = . 5 3. Assign a probability to each outcome: Pr : Ω → [ 0 , 1 ] . (a) Pr [ H ] = p , Pr [ T ] = 1 − p for some p ∈ [ 0 , 1 ] (b) Pr [ HH ] = Pr [ HT ] = Pr [ TH ] = Pr [ TT ] = 1 4 � 52 � (c) Pr [ A ♠ A ♦ A ♣ A ♥ K ♠ ] = ··· = 1 / 5
Probability Space: formalism. Ω is the sample space.
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point .
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point . (Also called an outcome .)
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point . (Also called an outcome .) Sample point ω has a probability Pr [ ω ] where
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point . (Also called an outcome .) Sample point ω has a probability Pr [ ω ] where ◮ 0 ≤ Pr [ ω ] ≤ 1;
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point . (Also called an outcome .) Sample point ω has a probability Pr [ ω ] where ◮ 0 ≤ Pr [ ω ] ≤ 1; ◮ ∑ ω ∈ Ω Pr [ ω ] = 1 .
Probability Space: formalism. Ω is the sample space. ω ∈ Ω is a sample point . (Also called an outcome .) Sample point ω has a probability Pr [ ω ] where ◮ 0 ≤ Pr [ ω ] ≤ 1; ◮ ∑ ω ∈ Ω Pr [ ω ] = 1 .
Probability Space: Formalism. In a uniform probability space each outcome ω is equally 1 probable: Pr [ ω ] = | Ω | for all ω ∈ Ω .
Probability Space: Formalism. In a uniform probability space each outcome ω is equally 1 probable: Pr [ ω ] = | Ω | for all ω ∈ Ω .
Probability Space: Formalism. In a uniform probability space each outcome ω is equally 1 probable: Pr [ ω ] = | Ω | for all ω ∈ Ω . Examples: ◮ Flipping two fair coins, dealing a poker hand are uniform probability spaces.
Probability Space: Formalism. In a uniform probability space each outcome ω is equally 1 probable: Pr [ ω ] = | Ω | for all ω ∈ Ω . Examples: ◮ Flipping two fair coins, dealing a poker hand are uniform probability spaces. ◮ Flipping a biased coin is not a uniform probability space.
Probability Space: Formalism Simplest physical model of a uniform probability space:
Probability Space: Formalism Simplest physical model of a uniform probability space:
Probability Space: Formalism Simplest physical model of a uniform probability space: A bag of identical balls, except for their color (or a label).
Probability Space: Formalism Simplest physical model of a uniform probability space: A bag of identical balls, except for their color (or a label). If the bag is well shaken, every ball is equally likely to be picked.
Probability Space: Formalism Simplest physical model of a non-uniform probability space:
Probability Space: Formalism Simplest physical model of a non-uniform probability space: p ω ω p 3 3 1 2 p 2 Fraction p 1 of circumference
Probability Space: Formalism Simplest physical model of a non-uniform probability space: p ω ω p 3 3 1 2 p 2 Fraction p 1 of circumference The roulette wheel stops in sector ω with probability p ω .
Probability Space: Formalism Simplest physical model of a non-uniform probability space: p ω ω p 3 3 1 2 p 2 Fraction p 1 of circumference The roulette wheel stops in sector ω with probability p ω . (Imagine a perfectly constructed wheel, that you spin hard enough, with low friction... .)
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads.
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition!
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E , is a subset of outcomes.
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E , is a subset of outcomes. Pr [ E ] = ∑ ω ∈ E Pr [ ω ] .
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E , is a subset of outcomes. Pr [ E ] = ∑ ω ∈ E Pr [ ω ] .
Probability of exactly one heads in two coin flips? Idea: Sum the probabilities of the outcomes with one heads. Leads to a definition! An event, E , is a subset of outcomes. Pr [ E ] = ∑ ω ∈ E Pr [ ω ] .
Event p ω ω p 3 3 1 2 p 1 p 2 E
Event p ω ω p 3 3 1 2 p 1 p 2 E Event E = ‘wheel stops in sector 1 or 2 or 3’
Event p ω ω p 3 3 1 2 p 1 p 2 E Event E = ‘wheel stops in sector 1 or 2 or 3’ = { 1 , 2 , 3 } .
Event p ω ω p 3 3 1 2 p 1 p 2 E Event E = ‘wheel stops in sector 1 or 2 or 3’ = { 1 , 2 , 3 } . Pr [ E ] = p 1 + p 2 + p 3
Probability of exactly one heads in two coin flips?
Probability of exactly one heads in two coin flips? Sample Space, Ω = { HH , HT , TH , TT } .
Probability of exactly one heads in two coin flips? Sample Space, Ω = { HH , HT , TH , TT } . Uniform probability space: Pr [ HH ] = Pr [ HT ] = Pr [ TH ] = Pr [ TT ] = 1 4 .
Probability of exactly one heads in two coin flips? Sample Space, Ω = { HH , HT , TH , TT } . Uniform probability space: Pr [ HH ] = Pr [ HT ] = Pr [ TH ] = Pr [ TT ] = 1 4 . Event, E , “exactly one heads”: { TH , HT } .
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