VIEWS: 50 PAGES: 29 POSTED ON: 3/14/2010 Public Domain
Probability Our Goals Develop a representation for illustrating the probabilities and outcomes of a particular scenario Practice this representation Investigate a complex scenario to illustrate the power of probability The Setting You are… Blindfolded, Led into a room, Instructed to throw a dart at the far wall until you are told to stop, Compute the probability of the possible outcomes if… You stop when you hit a balloon You stop when you hit a balloon You stop when you hit a balloon You stop when you hit a balloon You stop when you hit a balloon You stop when you hit a square You win You lose nothing $10 You win $8 You stop when you hit a square Win $10 Lose $3 Win $5 You stop when you hit a square +$25 -$15 +$40 -$10 -$20 +$10 +$5 -$15 Computing the Probability Heads on Tails on Flip 1 Flip 1 Heads on Flip 2 Tails on Flip 2 Computing the Probability Roll on First Die Pr(Sum>9) 1 2 3 4 5 6 Pr(Prod is even) 1 2 Roll on Second 3 Die 4 5 6 2 Draws with Replacement 3 Red Balls First Draw 4 Green Balls 1 Blue Balls Second Draw 2 Draws without Replacement 3 Red Balls First Draw 4 Green Balls 1 Blue Balls Second Draw Area Model Notice we can use area to see when we multiply or add probabilities We can even deal with dependent events As we get more familiar, we don’t need to be too accurate with scale- so long as we keep track of what the probabilities are An Application of Probabilities Your demographic has a low occurrence of AIDS, about 1 in 100,000. You take a test for AIDS that is 99.9% accurate, and it comes back positive Should you be worried? Why or why not? What if a second (independent) test came back positive? Scratch Work Generalizing We’ve now seen how the area model can be used to help compute the probabilities of a sample space (especially if the space is comprised of two distinct events) Heads Tails on on Flip 1 Flip 1 Heads on Flip 2 Tails on Flip 2 Generalizing So how might we try to adjust the area model so that it can succinctly represent more than two distinct events? Example: Use the area model to help you compute the probability of flipping atleast 2 heads in 3 coin flips Generalizing Example: Use the area model to help you compute the probability of flipping atleast 2 heads in 3 coin flips Heads Tails on on Flip 1 Flip 1 HT HH TT Heads on Flip 2 Tails on Flip 2 One way to generalize The first two flips HH TH TT H 3rd Flip T Pr(>1 H)= Pr(exactly 1 H)= Pr(<3 T)= Pr(3H)= Computing only one probability If you are interested in only one particular outcome, it is typically easier to compute only that probability and not model the entire scenario. Let us consider some examples… What is the probability that… Exactly 3 heads occur after 4 consecutive coin flips. – How many possible outcomes are there? – How many possible ways are there for there to be exactly 3 heads? Note: Counting can be reinterpreted as the number of choices one has (Recall the Cartesian product model of multiplication in 302A) What is the probability that… Exactly 2 heads occur after 5 consecutive coin flips. – How many possible outcomes are there? – How many possible ways are there for there to be exactly 2 heads? Birthday Problem Task: Compute the probability that at least two people in this room have the same birthday, Pr(shared). Birthday Problem Hint: Consider an simpler probability to compute that is related, namely: what is the probability that no one has a shared birthday. Pr(none shared) Question: If we can compute this probability, how do we find the original probability we were asked about? Summary Probability of an outcome in a scenario is the number of times the particular outcome can occur divided by the total number of all outcomes. Expected value represents the expected average value of the scenario as it is repeated many times. We can use an area model to represent the probabilities of a scenario Sometimes the probability that an outcome does not occur is easier to compute then the probability that it does. Homework 5 Exploration 7.18 in the Red Book – Parts I, II, V