Algebra 2 C Chapter 12 Probability and Statistics

Transcription

1 Algebra 2 C Chapter 12 Probability and Statistics Section 3 Probability fraction Probability is the ratio that measures the chances of the event occurring For example a coin toss only has 2 equally likely outcomes heads or tails so we say the PROBABILITY OF HEADS IS ½ while the PROBABILITY OF TAILS IS ½ odds verse probability is the odds slang for probability?? No, odds is not slang for probability. There is a difference. A probability is a number from 0 to 1 inclusive, usually expressed as a fraction, which is the ratio of the number of chances of a specific event to the total number of chances possible. For example, if I have 4 marbles in a jar, 3 red and 1 blue, then the probability of drawing the blue is 1/4. There is one chance of a blue marble and 4 total chances (marbles). Odds are expressed as the number of chances for (or against) versus the number of chances against (or for). So in the marble example the odds are 1:3 1 to 3 that you will draw a blue verse red marble 1

2 Probability is the ratio that measures the chances of the event occurring. # ways to " win" probability total # ways to " win" & " lose" Odds Emphasis on total is the ratio of one event compared to another odds # wins to win :# ways to lose Random means that the events are equally likely chances of occurring P(failure) = probability of failure P(success) = probability of success ie.) Coin toss P(heads) probability of heads P(F) + P(S) = 1 so the probability of any thing will always be less than one. David has a collection of 36 video games. 20 Action-Adventure games and 16 life simulation games. He is packing for a trip and is permitted by his parents to back 6 games. What is the probability if he selects 6 games at random that 3 will be action-adventure and 3 will be life simulations? Step 1 determine how many 6 game selections meet the conditions Step 2 use the fundamental counting principle to find s, the number of successes. Step 3 Find the total numbers s + f. of game selections. 2

3 David has a collection of 36 video games. 20 Action-Adventure games and 16 life simulation games. He is packing for a trip and is permitted by his parents to back 6 games. What is the probability if he selects 6 games at random that 3 will be action-adventure and 3 will be life simulations? Step 1 Determine how many 6 game selections meet the conditions C(20,3)= ways of selecting action games. Order does not matter C(16,3)= ways of selecting simulation games. Step 2 Use the fundamental counting principle to find s, the number of successes. 16! 20! * 13!3! 17!3! C(16,3) * C(20, 3) = = 638,400 David has a collection of 36 video games. 20 Action-Adventure games and 16 life simulation games. He is packing for a trip and is permitted by his parents to back 6 games. What is the probability if he selects 6 games at random that 3 will be actionadventure and 3 will be life simulations? C(16,3) * C(20, 3) = = 638,400 Step 3 Find the total numbers s + f. of game selections. 13! 20! * 13!3! 17!3! 36! C(36,6) = 30!6! = 1,947,792 = s + f 638,400 P( 3 action and 3 simulations) = = ,947,792 Example 2 A board game is played with lettered tiles. 56 are consonants and 42 are vowels. Each player must choose 7 tiles at the beginning of the game. What is the probability that a player selects 4 consonants and 3 vowels? C(56,4)* C(42,3) C(98,7) 4,216,489,200 13,834,413,152 31% 3

4 Ok now with permutations. Order matters Example 3 You have 5 books from 5 different classes: Algebra 2, Chemistry, History, English, and Spanish. You re going to put them on the shelf. If you pick them up at random and place them on the shelf. What is the probability that English, Spanish and Algebra 2 will be left most on the shelf, but not necessarily in that order? Example 3 You have 5 books from 5 different classes: Algebra 2, Chemistry, History, English, and Spanish. You re going to put them on the shelf. If you pick them up at random and place them on the shelf. What is the probability that English, Spanish and Algebra 2 will be left most on the shelf, but not necessarily in that order? Step 1 determine how many 5-book arrangements there are. Step 2 use the Fundamental Counting Principle to find s. Step 3 Find the total numbers s + f, of 5-book arrangements. P(English, Spanish, Algebra 2, followed by the other 2) Example 3 You have 5 books from 5 different classes: Algebra 2, Chemistry, History, English, and Spanish. You re going to put them on the shelf. If you pick them up at random and place them on the shelf. What is the probability that English, Spanish and Algebra 2 will be left most on the shelf, but not necessarily in that order? Step 1 determine how many 5-book arrangements there are. P(3,3) Place the 3 left most books P(2,2) Place the other 2 books Step 2 use the Fundamental Counting Principle to find s. P(3,3) * P(2, 2) = 3! * 2! = 12 Step 3 Find the total numbers s + f, of 5-book arrangements. P (5, 5) = 5! = 120 = s + f P(English, Spanish, Algebra 2, followed by the other 2) P(3,3)* P(2,2) P(5,5) % 4

5 Example 3 follow up.. You have 5 books from 5 different classes: Algebra 2, Chemistry, History, English, and Spanish. You re going to put them on the shelf. If you pick them up at random and place them on the shelf. What is the probability that English is the last book on the shelf? P (5, 1) = 5 Determine the probability. P(English, Spanish, Algebra 2, followed by the other 2) P(5,1) P(5,5) % Relative frequency histograms * used to help visualize probability distributions * tables or graphs of probabilities Relative frequency histograms * used to help visualize probability distributions * tables or graphs of probabilities 5

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