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Conditional probability

Ani Adhikari and Philip Stark Statistics 2.2X Lecture 2.1 1 / 6
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Multiplication rule for all A, B: P(A and B) = P(A) P(B|A)

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Multiplication rule for all A, B: P(A and B) = P(A) P(B|A)Rewrite:

Definition of conditional probability

P(B|A) =

P(Aand B)

P(A)

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P(B|A) =

P(Aand B)

P(A)

denominator: restricts the set of outcomes to what is givennumerator: the event of interest, only on the restricted set that is given

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P(B|A) =

P(Aand B)

P(A)


Exampleof use: Bayes Rule

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P(B|A) =

P(Aand B)

P(A)


Exampleof use: Bayes RuleBut many conditional probabilities can be found without this.

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P(B|A) =

P(Aand B)

P(A)


Exampleof use: Bayes RuleBut many conditional probabilities can be found without this.Example: Conditional probabilities for the second stage, given the result ofthe first stage

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Relation between draws

Two draws at random from R R R G G P(second draw is R )

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Two draws at random from R R R G G P(second draw is R ) = 3/5

regardless of whether the draws are with replacement or without

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P(second draw is R | first draw is R )

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P(second draw is R | first draw is R ) withreplacement withoutreplacement

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= 3/5 = 2/4

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= 3/5 = 2/4

P(second draw is R | first draw is G ) withreplacement withoutreplacement= 3/5 = 3/4

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= 3/5 = 2/4


independent

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= 3/5 = 2/4


independent dependent

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In practice

independent trials

tosses of a coin rolls of a die

drawswith replacement

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In practice

independent trials

tosses of a coin rolls of a die

drawswith replacement

trials that arenot independent (that is, dependent trials)

cards dealt from a deck

drawswithout replacement

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Independence

Rough definition

Two random quantities areindependentif knowing how one of themturned out does not change chances for the other.

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Independence

Rough definition


Independent events

Two events Aand Bare independent ifP(B|A) = P(B|not A) = P(B)

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Independence

Rough definition


Independent events


Recall multiplication rule for all A, B: P(A and B) = P(A) P(B|A)

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Independence

Rough definition


Independent events


Recall multiplication rule for all A, B: P(A and B) = P(A) P(B|A)Special case:

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Independence

Rough definition


Independent events


Recall multiplication rule for all A, B: P(A and B) = P(A) P(B|A)Special case:ifA and B are independent, P(Aand B) = P(A) P(B)

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Independence and multiplication

Two draws at random from R R R G G

P(both draws are R )

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P(both draws are R ) withreplacement withoutreplacement

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P(both draws are R ) withreplacement withoutreplacement= (3/5)(3/5) = (3/5)(2/4)

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Independence doesnt determinewhetheryou multiply; thatsdetermined byboth events have to happen.

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Independence doesnt determinewhetheryou multiply; thatsdetermined byboth events have to happen.

Independence affectswhatyou multiply.

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What to multiply

Example 1. A die is rolled twice. Can you find the chance that the firstroll shows an even number of spots and the second roll shows one spot?

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What to multiply

Example 1. A die is rolled twice. Can you find the chance that the firstroll shows an even number of spots and the second roll shows one spot?Answer. Yes;(3/6)(1/6)


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What to multiply


Example 2. A person is picked at random from a population. 50% of the population is male 10% of the population is left-handedCan you find the chance that the person is a left-handed male?


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What to multiply



Answer. No. You get stuck at

P(male)P(left handed|male) = 0.5P(left handed| male) = 0.5??


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What to multiply





The proportion of left-handers among males is not necessarily the same asthe overall proportion, 0.1.


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What to multiply





The proportion of left-handers among males is not necessarily the same asthe overall proportion, 0.1. Gender and handedness are dependent.

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