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Counting Principals Ex. Eight pieces of paper are numbered 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. A second piece of paper is drawn from the box, and its number is written down. How many different ways can a sum of 12 be obtained? Ex. Eight pieces of paper are numbered 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is NOT replaced in the box. A second piece of paper is drawn from the box, and its number is written down. How many different ways can a sum of 12 be obtained? Ex. How many different pairs of letters from the English alphabet are possible? Ex. How many different phone numbers are there in one area code? [Keep in mind that numbers can't start with 0 or 1.] A permutation is the rearrangement of elements. Ex. How many permutation are there of the letters A, B, C, D, E, and F? The number of permutations of n elements is n! Ex. How many distinguishable ways can the letters in BANANA be written? Ex. Eight horses are running in a race. How many different ways are there for the horses to take 1st, 2nd, and 3rd place? The number of permutations of n elements taken r at a time is n! nP r nr! In the last example, 8 ! 8 7 6 5 4 3 2 1 P 8 7 6 8 3 5 ! 5 4 3 2 1 A combination is a list of elements where order is not important. {A, B, C} and {B, C, A} are different permutations (where order matters) but the same combination (where order doesn't matter) The number of combinations of n elements taken r at a time is nCr Ex. A standard poker hand consists of five cards dealt from a deck of 52. How many different poker hands are possible? Ex. A 12-member swim team is being formed from 10 girls and 15 boys. If the team must consist of five girls and seven boys, how many different teams are possible? Practice Problems Section 8.6 Problems 5, 7, 13, 19, 39, 43, 55 Probability A sample space is a list of possible outcomes Ex. Find the sample space if a) one coin is tossed b) two coins are tossed The probability of an event is n u m b e r f a v o r a b l e P e v e n t n u m b e r p o s s i b l e Probabilities are always between 0 (impossible) and 1 (certain) Ex. Two coins are tossed. What is the possibility of both landing on heads? Ex. A card is drawn from a deck, what is the probability that it is an ace? Ex. Two dice are tossed. What is the probability that the total of the dice is 7? 52 Cards Face Cards ♣ ♠ ♦ ♥ 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 J J J J Q Q Q Q K K K K A A A A Ex. What is the probability of winning a lottery where a player chooses six numbers (in any order) between 1 and 41? P A o r B P A P B P A a n d B Ex. When drawing a card from a standard deck, what is the probability that the card is either a heart or a face card? ♣ ♠ ♦ ♥ 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 10 10 10 10 J J J J Q Q Q Q K K K K A A A A Two events are independent if their occurrences don't affect each other If A and B are independent, then P A a n d B P A P B Ex. A random number generator selects three integers from 1 to 20. What is the probability that all three numbers are less than or equal to 5? Remember, a probability of 1 is a sure thing. Sometimes it is easier to find the probability of the complement of an event. Ex. A manufacturer has determined that a machine averages one faulty unit for every 1000 that it produces. What is the probability that an order of 200 units will have one or more faulty units? Practice Problems Section 8.7 Problems 1, 11, 15, 17, 21, 45, 47, 48, 51