Counting Techniques (Dr. Monticino)

Overview  Why counting?  Counting techniques  Multiplication principle  Permutation  Combination  Examples  Probability examples

Why Counting?  Recall that if each outcome of an experiment is assumed to be equally likely, then the probability of an event is k/n  where k is the number of elements in the event and n is the number of elements in the sample space  So to calculate the probability of an event, we need to be able to count the number of elements in the event and in the sample space

Multiplication Principle  Multiplication principle. Suppose that an experiment can be regarded as a series of k sub- experiments. Such that the first sub-experiment has n 1 possible outcomes, the second sub- experiment has n 2 possible outcomes, and so on. Then the total number of outcomes in the main experiment is  n 1 x n 2 x... x n k  Examples  Flip a coin and roll a die  Roll 5 die; or roll a single die five times

Permutation  Factorial. n! (read “n factorial”) equals  Permutation. The number of ways to select r objects, in order, out of n objects equals

Examples  How many ways are there to do the following  Line up 10 people  Select a President, VP and Treasurer from a group of 10 people  Sit 5 men and 5 women in a row, alternating gender

Combination  Combination. The number of ways to select r objects out of n objects when order is not relevant equals

Examples  How many ways are there to do the following  Select 3 people from a group of 10  Select 7 people from a group of 10  Get exactly 5 heads out of 12 coin flips

Probability Examples  Select three people at random from a group of 5 women and and 5 men  What is the probability that all those selected are men?  What is the probability that at least one women is chosen?  What is the probability that at least two women are chosen?

Probability Examples  Flip a fair coin 3 times  What is the probability that 3 heads come up?  What is the probability that at least 1 tail occurs?  What is the probability that exactly 2 tails occur?  What is the probability that at least 2 tails occur?

Probability Examples  Play roulette 3 times  What is the probability that red comes up every time?  What is the probability that black comes up at least once?  What is the probability that black comes up exactly two times?  What is the probability that black comes up at least two times?

Probability Examples  Flip a fair coin 10 times  What is the probability that 10 heads come up?  What is the probability that at least 1 tail occurs?  What is the probability that exactly 8 tails occur?  What is the probability that at least 8 tails occur?

Probability Examples  Play roulette 20 times  What is the probability that red comes up every time?  What is the probability that black comes up at least once?  What is the probability that black comes up exactly 18 times?  What is the probability that black comes up at least 18 times?

Probability Examples  Roll a fair die 5 times  What is the probability that an ace comes up all five times?  What is the probability that an ace occurs at least once?  What is the probability that an ace occurs exactly 3 times?  What is the probability that an ace occurs at least 3 times?

Probability Examples  To win the jackpot in Lotto Texas you need to match all six of the numbers drawn (5 numbers are selected from numbers 1 to 44 and the sixth is selected separately from 1 to 44)  What is the probability of winning if you buy one ticket?  What is the probability of winning if you buy five tickets?  Is it better to buy five tickets in one Lotto drawing or a single ticket in five successive Lotto games?

Assignment Sheet  Read Chapter 15 carefully  Redo all problems from lecture  Not to turn in… (Dr. Monticino)