Probability and Statistics Unit 8

For lunch, in how many different ways could you choose a sandwich, side dish, and dessert? Think about it

The Fundamental Counting Principle If an event M that can occur in m ways is followed by event N that can occur in n ways, then event M followed by event N can occur in 𝑚∙𝑛 ways. Example: 3 pants and 2 shirts give 3∙2=6 possible outfits. The Fundamental Counting Principle

In 1966, one type of Maryland license plate had two letters followed by four digits. How many of this type of license plate were possible? How many license plates would be possible if the state did not allow repeated numbers or letters? Examples

A permutation is an arrangement of items in a particular order A permutation is an arrangement of items in a particular order. Suppose you wanted to find the number of ways to order three items. There are 3 ways to choose the first, 2 ways to choose the second, and only 1 way to choose the last. 3∙2∙1=6 permutations Permutation

Using factorial notation, you can write 3∙ 2∙1 as 3 Using factorial notation, you can write 3∙ 2∙1 as 3!, read “3 factorial.” For any positive integer n, n factorial is 𝑛!=𝑛 𝑛−1 ∙ . . . ∙3∙2∙1. Zero factorial is 0!=1. Factorial Notation

How many ways can you file 12 folders, one after another, in a drawer How many ways can you file 12 folders, one after another, in a drawer? How many ways can you hang 8 shirts on hangers in a closet? Example

Number of Permutations The number of permutations of n items of a set arranged r items at a time is 𝑛𝑃𝑟 = 𝑛! 𝑛−𝑟 ! for 0≤𝑟≤𝑛 Example: 10𝑃4= 10! 10−4 ! = 10! 6! =5040 Number of Permutations

Ten students are in a race Ten students are in a race. First, second, and third places will win medals. In how many ways can 10 runners finish first, second, and third (no ties allowed) Example

What if in the last example the first three runners would get to go to the state championship. Then order wouldn’t matter. A selection in which order doesn’t matter is called a combination. Combinations

Number of Combinations The number of combinations of n items of a set chosen r items at a time is 𝑛𝐶𝑟= 𝑛! 𝑟! 𝑛−𝑟 ! 𝑓𝑜𝑟 0≤𝑟≤𝑛 Example: 5𝐶3 = 5! 3! 5−3 ! = 5! 3!∙2! = 120 6∙2 =10 Number of Combinations

How many committees of 4 could you choose from 13 people? Example

An art teacher divides his class into five groups An art teacher divides his class into five groups. Each group submits one sketch. She will select 3 of the sketches to display. In how many different ways can she select the sketches? Example

You will draw winners from a total of 25 tickets in a raffle You will draw winners from a total of 25 tickets in a raffle. The first ticket wins $100. The second ticket wins $50. The third ticket wins $10. In how many different ways can you draw the three winning tickets? Example

Experimental Probability Experimental probability of an event: 𝑃 𝑒𝑣𝑒𝑛𝑡 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠 The probability of an impossible event is 0 (or 0%). The probability of a certain event is 1 (or 100%). Otherwise, the probability of an event is a number between 0 and 1 (or a percent between 0% and 100%). Experimental Probability

On Thursday, there were 68 cars in the teacher’s parking lot On Thursday, there were 68 cars in the teacher’s parking lot. Fourteen of the cars were SUV’s. What is the experimental probability that a vehicle picked as random is an SUV? What is the experimental probability that the vehicle is NOT an SUV? Example

A baseball player got a hit in 20 of his last 50 times at bat A baseball player got a hit in 20 of his last 50 times at bat. What is the experimental probability that he will get a hit in his next at bat? Example

On a multiple-choice test, each item has 4 choices, but only one choice is correct. How can you simulate guessing the answers? What is the probability that you will pass the test by guessing at least 6 of 10 answers correctly? Simulation

The set of all possible outcomes to an experiment or activity is call the sample space. When each outcome in a sample space has the same chance of occurring, the outcomes are equally likely outcomes. When outcomes are equally likely, you can calculate the theoretical probability of the event. Terms to Know

Theoretical Probability The theoretical probability of event A is 𝑃 𝐴 = 𝑚 𝑛 Sample space: n outcomes Event A: m outcomes Theoretical Probability

What is the theoretical probability of getting a 5 on one roll of a standard number cube? What is the theoretical probability of getting a sum of 5 on one roll of two standard number cubes? Examples

What is the probability of being dealt exactly two 7’s in a 5-card hand from a standard 52- card deck? Using Combinatorics

What is the probability of being dealt 3 5’s and 2 aces? Another one. . .

Geometric Probability What is the probability that a dart thrown at this board lands in the bullseye? 20 in 3in Geometric Probability

What is the probability that a checker thrown on top of this board lands on a black square? Another geometric

What is the probability a quarterback will complete his next pass if he has completed 30 of his last 40 passes? Example

What is the probability a quarterback will complete his next pass if he has completed 36 of his last 45 passes? Example

Find the probability of each event when rolling a standard die Find the probability of each event when rolling a standard die. P(3) P(2 or 4) Example

Probability of Multiple Events

Dependent or Independent? To find the probability of two events occurring together, you have to decide whether one event occurring affects the other event. Dependent Events—the outcome of a second event depends on what happens first. Independent Events—the first outcome does not affect the second outcome Dependent or Independent?

Dependent or Independent? Roll a number cube. Then spin a spinner. Pick one flash card, then another from a stack of 30 flash cards. Draw one marble from a bag, then replace it and draw another. Draw one marble from a bag, then draw another without replacing. Dependent or Independent?

If A and B are independent events, then 𝑃 𝐴 𝑎𝑛𝑑 𝐵 =𝑃(𝐴)∙𝑃(𝐵) Example: At a picnic there are 10 diet drinks and 5 regular drinks. There are also 8 bags of barbecue chips and 12 bags of regular chips. If you grab a drink and bag of chips, what is the probability that you get a diet drink and regular chips? Probability of A and B

Mutually Exclusive Events Two events that cannot happen at the same time are called mutually exclusive. You roll a standard die. Are these events mutually exclusive? Rolling a 2 and a 3 Rolling an even number and a multiple of 3 Rolling an even number and a prime number Mutually Exclusive Events

𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃 𝐵 −𝑃 𝐴 𝑎𝑛𝑑 𝐵 If A and B are mutually exclusive events, then 𝑃 𝐴 𝑜𝑟 𝐵 =𝑃 𝐴 +𝑃(𝐵) Probability of A or B

A student can take one foreign language each term A student can take one foreign language each term. About 37% of students take Spanish. About 15% of students take French. What is the probability that a student chosen at random is taking Spanish or French? Examples

Suppose you reach into this dish and select a token Suppose you reach into this dish and select a token. What is the probability that the token is round or green? Example