AMS 310 Lecture 4

AMS 310 Lecture 4 - Conditional Probability 1. Independent...

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Conditional Probability 1. Independent Events Informally we say that two events A and B are called independent if the occurrence of A does not affect the probability of B occurring. Examples Rolling a pair of dice repeatedly Coin flipping Basketball free throws The price of gold and the price of tech stocks is almost independent Theorem Two events A and B are independent, if and only if P(A B) = P(A)P(B) Examples a) A coin is tossed and a die is rolled. What is the probability of getting a head and a 4? Solution P(H 4) = P(H)P(4) = (1/2)(1/6) = 1/12 b) A fair coin is flipped three times. What is the probability of getting three heads? c) It is estimated that 9% of men are color blind. If 3 men are selected at random, find the probability that all of them are color blind.
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When the occurrence of an event does affect the occurrence of another event, we say that the events are dependent . Examples a) Two cards are drawn from a deck and the first one is not replaced before the second one is drawn. b) The interest rate and the price of bonds (When the interest rate goes up, the price of bonds goes down) 2. Conditional Probability The conditional probability of A given that B has occurred is denoted by P(A│B). Let S = {1,2,3, …, 25} Let B = {1,2,3,4,5,6,7,8,9,10} Let A = {2,4,6,8,10} Suppose that a number is chosen randomly from S. Then P(A) = 525 and P(B) = 1025 . What is P(A│B) ? What is P(A B)? What is P(B│A) ?
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3. Formula for Conditional Probability: If A and B are any events, then P(A│B) = ( ∩ ) ( ) P A B P B In the above example, P(A B) = 525 and P(B) = 1025 . Using the formula we get P(A│B) = ( ∩ ) ( )= = P A B P B 5251025 = . 510 5 Example Blood types are often grouped as follows. O A B AB Rh+ 37% 34% 10% 4% Rh- 6% 6% 2% 1% Find the probability that a person is: a) O negative (universal donor) b) Type O given that the person is Rh+ c) Has A+ or AB- d) Has Rh- given that the person has type B e) Type A given that the person is Rh- The formula for conditional probability can also be used to find P(A B).
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By solving for P(A B) we get P(A B) = P(B) P(A│B). This could also give us P(A B) = P(A) P(B│A). Example Two cards are drawn at random from an ordinary deck of 52. What is the probability of getting two aces if a) the first card is replaced before the second is drawn b) the first card is not replaced before the second is drawn Solution Let A be the event an ace is chosen on the first draw Let B be the event an ace is chosen on the second draw
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This note was uploaded on 03/15/2011 for the course AMS 310 taught by Professor Mendell during the Spring '08 term at SUNY Stony Brook.

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AMS 310 Lecture 4 - Conditional Probability 1. Independent...

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