DARTMOUTH COSC 030 - 09condprob (4 pages)

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Conditioning Pick uniform random card from standard deck CS 30 Discrete Mathematics Amit Chakrabarti Conditional Probability Independent Events Four Step Method Now with Conditioning Take S 2 2 7 Q A Pr x 1 52 for each x S Event A the card has a prime number on it Event R the card is red Event B the card has either a 5 or a 6 on it Event Q the card is either a Queen or a King Pr R 26 52 1 2 Pr B 8 52 2 13 Pr Q 8 52 2 13 Under the condition that A occurs Under the condition that A occurs New probability function Pr x 0 if x A Pr x if x A Normalization Pr x Pr x Pr A so that all Pr values add up to 1 Pr R A Pr R Pr R A Pr R A 0 Pr R A Pr A Pr R A Pr A 8 52 16 52 1 2 Pr B A Pr B A Pr A 4 52 16 52 1 4 Pick uniform random card from standard deck Take S 2 2 7 Q A Pr x 1 52 for each x S Event A the card has a prime number on it x S x has a prime number on it Event R the card is red Event B the card has either a 5 or a 6 on it Pr R A Pr R A Pr A 8 52 16 52 1 2 Pr B A Pr B A Pr A 4 52 16 52 1 4 Pr Q A Pr Q A Pr A 0 16 52 0 R and A are independent Pr R A Pr R Dartmouth is playing Team H in a 3 game series Dartmouth wins Game 1 with probability 1 2 Upon winning a game Dartmouth wins next game with probability 2 3 Upon losing a game they win the next game with probability 1 3 Pr Dartmouth wins series Dartmouth won Game 1 1 Find the sample space S WW WLW LL LWL LWW WLL 2 Deline events of interest F Dartmouth wins Game 1 WW WLW WLL T Dartmouth wins series WW WLW LWW 3 Determine outcome probabilities Tree diagram on whiteboard 4 Compute event probabilities Four Step Method Now with Conditioning Pr Dartmouth wins 3 game series Dartmouth won Game 1 Dartmouth wins Game 1 with probability 1 2 After winning win next w p 2 3 after losing win next w p 1 3 1 Find the sample space A Serious Example Medical Testing Mammogram 10 false negative 5 false positive Middle aged women no family history of cancer 1 chance of cancer If one in this cohort tests positive what s the chance they have