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Chapter 14: From Randomness to Probability Sami Sahnoune Amin Henini Probability ● probability - an event’s long-run relative frequency ● each trial generates an outcome ● the probability allows us to see general outcomes that would happen in the long run ● independent trial - outcome of one trial doesn’t influence outcome of another ● Law of Large Numbers (LLN) - long-run relative frequency gets closer and closer to true relative frequency as the number of trials increases Probability Vocabulary ● Random Phenomenon - A phenomenon is random if we know what outcomes could happen, but not which particular values will happen. ● Probability - The probability of an event is a number between 0 and 1 that reports the likelihood of the event’s occurrence. P(A) represents the probability of event A. ● Trial - A single attempt or realization of a random phenomena. ● Outcome - The outcome of a trial is the value measured, observed, or reported for an individual instance of that trial. ● Event- A collection of outcomes. ● Independence - Two events are independent if knowing whether one event occurs does not alter the probability that the other event occurs. ● Disjoint - Two events are disjoint if they share no outcomes in common. Disjoint events are also called “mutually exclusive”. Probability and Rules ● probability must be between 0 and 1, inclusive o A probability of 0 indicates impossibility. o A probability of 1 indicates certainty. ● “Something Has to Happen Rule”- Sum of all probabilities must equal one ● Complement Rule- The probability an event will happen is 1 minus the probability it won’t happen Probability and Rules ● Addition Rule- If A and B are disjoint events, then the probability of A or B is P(A) + P(B) ● Multiplication Rule- If A and B are independent events, then the probability of A and B is P(A) * P(B) Probability Notation ● P( A ∪ B) or P(A or B) ● P(A ∩ B) or P(A and B) ● An important thing to remember is that working with the complement of the event we care about can often be easier. Things To Avoid ● Make sure that your probabilities end up adding to one. ● Don’t add probabilities together if they aren’t disjoint. ● Don’t multiply probabilities together if they aren’t independent. ● Don’t mix up disjoint and independent events disjoint events can’t be independent. Summary ● Remember that the probability of one event or multiple events must end up equaling one. If not, then the probabilities that were assigned to the outcomes aren’t legitimate. ● Make sure you remember that LLN only applies to long-run relative frequencies. ● The four rules used to calculate probabilities are: 1. Multiplication Rule 2. Addition Rule 3. “Something Has to Happen” Rule 4. Complement Rule ● These rules can be combined in various ways in order to find the probabilities of complex events. Problems 25. Disjoint or Independent? The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&M’s, red another 20% and orange, blue, and green each made up 10%. The rest were brown. a. If you draw one M&M, are the events of getting a red one and getting an orange one disjoint or independent or neither? For one draw, the events of getting a red M&M and getting an orange M&M are disjoint events, as you cannot draw both red and orange. b. If you draw two M&M’s one after the other, are the events of getting a red on the first and a red on the second disjoint, independent, or neither? They are independent, since drawing red the first time does not mean you will draw a red one the second time. c. Can disjoint events ever be independent? Explain. Disjoint events can never be independent because in a disjoint event there can only be one outcome. This will affect the probability of all the other outcomes, something which doesn’t happen independent events. Problems 27. Dice. You roll a fair die three times. What is the probability that: a) you roll all 6's? P(6) = ⅙ , so P(all 6’s) = (⅙)(⅙)(⅙) = 0.005 b) you roll all odd numbers? P(odd) = P(1 ∪ 3 ∪ 5) = ½, so P(all odd) = (½)(½)(½) = 0.125 c) none of your rolls gets a number divisible by 3? P(not divisible by 3) = P(1 or 2 or 4 or 5) = 4/6 = ⅔ P(none divisible by 3) = (⅔)(⅔)(⅔) = 0.296 d) you roll at least one 5? P(at least one 5) = 1 – P(no 5’s) = 1 - (⅚)(⅚)(⅚) = 0.421 e) the numbers you roll are not all 5's? P(not all 5’s) = 1 – P(all 5’s) = 1 - (⅙)(⅙)(⅙) = 0.995