Main Concepts: List all possible outcomes of a chance process in a systematic way Design simulations and use them to estimate probabilities Use the Addition rule to compute probability that event A or event B (or both) occurs Use the Multiplication rule to compute probability that event A and event B both occur. Compute Conditional probabilities, the probability that event B occurs given that event A occurs (the most difficult)
An event is a set of possible outcomes from a random situation: rolling dice, drawing a card, pulling a marble from a bag, result of some type of spinner… Probability is a number between 0 and 1 (or between 0% and 100%. Something that is certain to occur has a probability of 1. Something that will not occur has a probability of 0.
The probability that event A occurs is denoted by P(A). The probability that event A does not occur is then, P(not A) = 1 – P(A) “not A” is also called the compliment of A. If you have a list of all possible outcomes and all outcomes are equally likely, the probability of a specific outcome is: the number of outcomes of that event / the total number of equally likely outcomes
Page 288: Identifying Tap or Bottled water: Probability Distribution: a chart or graph showing the possible values from the random process and the probabilities of each. The sum of the probabilities must be….?
A Sample Space for a chance process is a complete list of disjoint outcomes. All of the outcomes in a sample space must have a total probability equal to 1. Disjoint: two different outcomes can’t occur in the same opportunity. (AKA: mutually exclusive)
Think of a Tree Diagram which maps out the possibilities.Tree Diagram If you have n 1 possible outcomes for stage 1 and n 2 possible outcomes for stage 2, n 3 possible for stage 3 and so on, then the total possible for the stages together is n 1 n 2 n 3. 2 people guessing T or B. 3 people guessing T or B.
6 possibilities on each. n 1 =6, and n 2 =6 So there are 6*6 = 36 equally likely outcomes. Notice for example 3,4 and 4,3 have the same result, but were different in how they happened, so they are considered two different outcomes. However 3,3 is the same as 3,3 so those are not two different outcomes.
In random sampling, the larger the sample, the closer the proportion of successes in the sample tends to be to the proportion in the population. ie: Think of flipping a coin. If you flipped the coin twice, would you expect Heads 50% of the time (would you be surprised if you got two Heads)? If you rolled the dice 5000 times, would you expect Heads approximately 50% of the time (would you be surprised if you got 5000 Heads)? This is an example of the Law of Large Numbers. You are really comparing theoretical probability to experimental probability.
A sample space together with an assignment of probabilities. Each will have probability between 0-1. All together they will sum to 1. Homework: p296 P1-P9, E1,2,3,7,9,13