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M2 U6 Probability Flowchart

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  Unit 6 Probability Flowchart This flowchart is designed to help organize the large variety of types of Probability, Outcome and Odds problems. Use the examples  below the chart to help sort out how problems fit into each category. Probability, Odds or Outcome    –   These are the three main categories of problems you will encounter. Probability    –   A probability problem ask you to find the probability or likelihood of something happening.  Favorable OutcomesTotal Outcomes   Odds    –   Odds problems ask you to find the odds in favor or against a particular event happening.  Favorable OutcomesUnfavorable Outcomes   Outcome    –   These problems ask you to find the number of ways a particular event can happen. # of Outcomes   PermutationOrder withina GroupPermutationOrder withina GroupSimpleMenuProblemsCombinationGroupswithoutorder Binomial EventP=1/2 P-1/2 CompoundEventsMenuProblemsCombinationGroupingwithoutorder Probability(Likelihood)Favorable/TotalOdds Not Probability Favorable/UnfavorablePermutationOrder withina GroupPermutationOrder withina GroupSimpleMenuProblemsCombinationGroupswithoutorder CompoundEventsMenuProblemsCombinationGroupingwithoutorder Outcomes# of wayssomethingcan happenProbability  Compound Events    –   License plate/Phone number and Menu problems, with and   without repetition (Hint: the word and   should translate to multiply ; if you see the word or,  think add  ) i.e.: How many ways can a meal be assembled if there are 5 choices for appetizers, 8 for main course, 2 salads, 3 soups and 7 desserts? (Include one choice from each category.) Outcomes  = 5(8)(2)(3)(7) = 1680 ways of making a meal. i.e.: When drawing two cards, (standard poker deck) what is the probability of choosing a Jack and a 7? (without replacement) Probability  = 45245116   2652   Permutation    –   Grouping problems where order is important. Political offices / running races and problems without repetition i.e.: How many different 7 letter “words” can be made from all letters? (of the alphabet)   Outcomes  = 26  P  7  = 26 25 24 23 22 21 20  = 3,315,312,000 ways. i.e.: How many different ways can the letters in the word MONTANAN be written? Outcomes  = 8!3! 2!  or 8 7 6 5 4 3 2 1(3 2 1)(2 1)  = 3360 (The values in the denominator represent the repetition of the letters N and A.) Permutation Probability i.e.: There are 25 people running a race; 11 women and 14 men. The top three  places are recognized. What is the probability that the three   places will be all women? Probability  = 11  P  325  P  3 11 10 9   25 24 230.0717    Combination    –   Grouping problems where order is not   important. i.e.: How many different ways can a committee of 5 be selected from a 12 person board of directors? Outcomes  = 12 C 5  = 12! 12 11 10 9 8 7 6 5 4 3 2 1792(12 5)!5! (7 6 5 4 3 2 1)(5 4 3 2 1)   Combination Probability i.e.: There are 5 green marbles, 4 red and 7 blue in a bag. If you take 4 marbles out of the bag, what is the  probability that all 4 are blue ones? Probability  = 7 C  416 C  4   0.0192   Compound Combination Probability (use the set-up from the problem listed in Combination Probability ) i.e.: If the top five   places are recognized, what is the probability that the  five   places will be 3 women and 2 man? Probability  = 11 3 14 225 5 0.2826 C C C   Binomial Event    –   Multiple coin flips or t rue/false questions or wins/loss problems. (Individual probability isn’t necessarily = 0.5.)  i.e.: For a team with whose probability of winning a game is 0.75, what is the probability of the team winning 7 of the next 10 games? Probability (winning 7 of the next 10 games)  = 10 C 7  * (.75) (.75) (.75) (.75) (.75) (.75) (.75) (.25) (.25) (.25) = 0.2503 Or Probability (winning 7 of the next 10 games)  = 10 C 7  * (.75) 7  (.25) 3 = 0.2503    In the first version of the solution, the combination at the  beginning (It can be found in Pascal’s Triangle 10 th  row, 7 th  value) shows all the orders in which the games can be won. The seven .75’s represent the seven wins and the three .25’s represent the three l osses. The second version simplifies the process by using powers for the wins and losses. Either method results in the same answer. Combinations and Pascal’s Triangle  At Least and At Most can complicate these problems. i.e.: When a problem asks something like: In a true/false quiz with 7 questions what is the probability that at least 5 of the 7 questions are answered correctly by guessing? To answer this problem, you must find the probability that: exactly 5 of 7 are answered correctly exactly 6 of 7 are answered correctly exactly 7 of 7 are answered correctly Then add the three probabilities together. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Several number patterns are found within the rows of Pascal’s Triangle.  If you add each row, you will find that the sums are powers of two. There are a variety of patterns in the diagonals Each term within the triangle is a Combination (remember to begin with row zero and zero at the beginning of each row) i.e.: 7 C 2  is 21. To find it, go to the seventh row, the second term. For 8 C 6 , you find the eighth row and the 6 th  term, which is 28. This can be helpful for finding the values in a binomial event.
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