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03_probability1

# 03_probability1 - Class details Probability I 9.07...

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Probability I 9.07 2/10/2004 Probability and gambling De Mere: “Which is more likely, rolling at least one 6 in 4 rolls of a single die, or rolling at least one double 6 in 24 rolls of a pair of dice?” De Mere reasoned they should be the same: Chance of one 6 in one roll = 1/6 Average number in 4 rolls = 4·(1/6) = 2/3 Chance of one double 6 in one roll = 1/36 Average number in 24 rolls = 24·(1/36) = 2/3 Why, then, did it seem like he lost more often with the second gamble? He asked his friend Pascal, and Pascal & Fermat worked out the theory of probability. Class details Reminder: HW 1 due on Friday. HW2 is now on the web. It’s due Friday of next week. Readings in Probability now on the web. Reminder: Office hours today, 3-4 pm Basic definitions Random experiment = observing the outcome of a chance event. Elementary outcome = any possible result of the random experiment = O i Sample space = the set of all elementary outcomes. 1

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Example sample spaces Sample space for a pair of dice Tossing a single coin: –{H , T} Tossing two coins: – {HH, TH, HT, TT} One roll of a single die: Each pair is an elementary outcome. Fair coin or die Properties of probabilities For a fair coin or die, the elementary •P (O i ) 0 outcomes have equal probability – Negative probabilities are meaningless – P(H) = P(T) = 0.5 • The total probability of the sample space – P(1 spot) = P(2) = P(3) = P(4) = P(5) = P(6) = must equal 1. – If you roll the die, one of the elementary Of course, the coin or die might not be fair outcomes must occur. P = .15 .10 .25 .15 .15 .20 2 1/6
3 How do we decide what these probabilities are? • 1. Probability = event’s relative frequency in the population . – Look at every member of the population, and record the relative frequency of each event. – Often you simply can’t do this. • 2. Estimate probability based on the relative frequency in a (large) sample. – Not perfect, but feasible. • 3. Classical probability theory: probability based on an assumption that the game is fair. – E.G. heads and tails equally likely. – Similarly, might otherwise have a theoretical model for the probabilities. Events • An event is a set of elementary outcomes. • The probability of an event is the sum of the probabilities of the elementary outcomes. • E.G. tossing a pair of dice: Event A: Dice sum to 3 Event B: Dice sum to 6

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4 Event C: White die = 1 Event D: Black die = 1 Combining events •E AND F: both event E and event F occur OR F: either event E occurs, or event F does, or both NOT E: event E does not occur C OR D: W=1 OR B=1
5 The addition rule • P(W=1) = 6/36 • P(B=1) = 6/36 •P (W=1 o r B=1 ) P(W=1) + P(B=1)

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03_probability1 - Class details Probability I 9.07...

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