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Intro to Probability Objectives: To evaluate the big idea in Probability: chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run (Law of Large Numbers). To use Venn diagrams as a visual aid for understanding concepts in context Warm-up: 1. If you have 6 different books to place on a bookshelf, how many different arrangements are possible? 654321=720 2. How many different combinations exist for your locker? 503 =125000 3. How many different locker combinations would there be if you could only use each number once? 504948 =117600 Questions in modern day probability: Should I spend money on a warranty for my new Ipod? If I test positive for a rare blood disease, does this mean that I definitely have this disease? Can we determine the chances of a child having a psychological disorder based on heredity? Random Phenomenon A phenomenon is random if individual outcomes are uncertain, but there is a predictable distribution of outcomes over many repetitions. Experimental versus Theoretical Probability Probability The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. (Probability is basically long term relative frequency) Sample Space (S) The set of all possible outcomes for some type of random phenomenon Examples: Coin Toss S = {H, T} Fair die S = {1, 2, 3, 4, 5, 6} Toss a coin twice S = {HH, TT, HT, TH} How many outcomes are there in the sample space for rolling two dice? 36 Event An event is any outcome or a set of outcomes of a random phenomenon. An event is basically a subset of the sample space. Examples: Rolling a Prime # A = {2, 3, 5} Rolling a Prime # or an even # B = {2, 3, 4, 5, 6} Complement c E Consists of all outcomes that are not in the event Example: Rolling an even # E={2,4,6} Complement: Not rolling an even # EC={1,3,5} Union E AB the event A or B happening consists of all outcomes that are in at least one of the two events Ex. Rolling a prime # or even number W ={2,3,4,5,6} Intersection E A B the event A and B happening consists of all outcomes that are in both events Example: Drawing a red card and a “2” L = {2 of hearts, 2 of diamonds} Mutually Exclusive (disjoint) two events have no outcomes in common Example: The event of rolling an even # is disjoint from the event of rolling an odd # Probability model Mathematical description of a random phenomenon consisting of two parts 1. A sample space (S) 2. A method of assigning probabilities to each event We will focus on part 1 today… Tree Diagram It is very important to check that we have not overlooked any possible outcome. One visual method of checking this is making use of a tree diagram. Ex. Flip a coin, then roll a die S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6} Multiplication Principle If you can do one task in x number of ways and a second task in y number of ways, then the sample space of both task can be shown with x ● y possible outcomes. Example: Tossing a coin – two possible outcomes Rolling a die – six possible outcomes Tossing a coin, then rolling a die: 2 ● 6 = 12 possible outcomes Venn Diagrams Used to display relationships between events Helpful in calculating probabilities Venn diagram - Complement of A AA Venn diagram - A or B A B Venn diagram - A and B A B Venn diagram - disjoint events A B Independence The outcome of one trial must not influence the outcome of another trial. This is a major concept in statistics that is often neglected in the design and data collection process. We will look at independence both logically and mathematically in this course. Closing: 1. In your own words, describe random phenomena. 1. Write down the symbol, key word, and visual display for Union 1. Write down the symbol, key word, and visual display for Intersection