Data Analysis and Probability Presenters Aaron Brittain Adem Meta

MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes. MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes. MA.9.5.I: Design an experiment to test a theoretical probability, and record and explain results. MA.9.5.I: Design an experiment to test a theoretical probability, and record and explain results. MA.9.5.J: Compute probabilities of compound events, independent events, and simple dependent events. MA.9.5.J: Compute probabilities of compound events, independent events, and simple dependent events. MA.9.5.K: Make predictions based on theoretical probabilities and experimental results. MA.9.5.K: Make predictions based on theoretical probabilities and experimental results.

INDEPENDENT EVENTS: when the occurrence of one event has no effect on the probability that a second event will occur.

DEPENDENT EVENTS: is the occurrence of one event does have an effect on the probability that a second event will occur.

PROBABILITY OF TWO INDEPENDENT EVENTS P(A and B) = P(A) x P(B) EX: Find the probability of flipping a coin and getting heads and then rolling a 6 on a number cubed. 1. Identify the event as independent or dependent. 2. Find the probability of each event P(heads) = ½

P(6 on a number cube) = 1/6 3. multiply the two results and simplify your fraction if necessary ½ x 1/6 = 1/12

Holt 10.7 Additional Algebra Lab Pp MA.9.5.J: Compute probabilities of compound events, independent events, and simple dependent events. MA.9.5.J: Compute probabilities of compound events, independent events, and simple dependent events.

Holt 10.8 Algebra Lab Pp MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes. MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes.

Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. "The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be exactly So, in Mathematics we use more precise language: If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. So, we should really call this a "Permutation Lock"! In other words:

A Permutation is an ordered Combination. To help you to remember, think "Permutation... Position" Permutations There are basically two types of permutation: Repetition is Allowed: such as the lock above. It could be "333". No Repetition: for example the first three people in a running race. You can't be first and second.

1. Permutations with Repetition These are the easiest to calculate. When you have n things to choose from... you have n choices each time! When choosing r of them, the permutations are: n × n ×... (r times) (In other words, there are n possibilities for the first choice, THEN there are n possibilities for the second choice, and so on, multplying each time.) Which is easier to write down using an exponent of r:exponent n × n ×... (r times) = nPr Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them: 10 × 10 ×... (3 times) = 10^3 = 1,000 permutations So, the formula is simply: nPr where n is the number of things to choose from, and you choose r of them (Repetition allowed, order matters)

2. Permutations without Repetition In this case, you have to reduce the number of available choices each time. For example, what order could 16 pool balls be in? After choosing, say, number "14" you can't choose it again. So, your first choice would have 16 possibilites, and your next choice would then have 15 possibilities, then 14, 13, etc. And the total permutations would be: 16 × 15 × 14 × 13 ×... = 20,922,789,888,000 But maybe you don't want to choose them all, just 3 of them, so that would be only: 16 × 15 × 14 = 3,360

One class has 20 students. If the math teacher wants to create groups of 3 students, how many possibilities are there? In the same class, if they were to elect in each group one president, one vice-president, and one secretary, how many possible groups are there?

Ohio license plates have four numbers and three letters. How many plates can be produced? If a group of five people are in the room at the same time, how many handshakes are there if the all greet each other?

Probability Mass Function The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials.

The formula for the binomial probability mass function is:

On the 2009 OGT Math Test, there were 31 multiple choice questions. (4 possible answers for each question) What is the probability that a student to get 20 correct answers by random selection ? What is the probability to get at least 27 correct answers? What is the probability to get at most 4 correct answers?

If a person toss a coin three times in a row, find the number possibilities in the sample space. (draw the tree) What is the probability to get three tails? What is the probability to get at least two heads? What is the probability to get at most one tail?

Create a sample space for the total number of outcomes when rolling two number cubes Find the probability of the sum of two numbers equaling six MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes. MA.9.5.H: Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes.

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