Chapter 13 Probability and Measurement

Chapter 13 – Probability and Measurement Section 1 – Representing Sample Spaces  I can represent sample spaces.  I can use the Fundamental Counting Principal to count outcomes.

 An experiment is a situation involving chance that leads to results called outcomes. (ex: tossing a coin)  An outcome is the result of a single performance or trial of an experiment. (ex: landing on heads or tails)  An event is one or more outcomes of an experiment. (ex: the coin landing on tails)  Sample space is the set of all possible outcomes. Can be represented using a list, table or tree diagram  Fundamental Counting Principle – the number of possible outcomes (page 901)

1. One red token and one black token are placed in a bag. A token is drawn and the color is recorded. It is then returned to the bag and a second draw is made. Represent the sample space for this experiment by making an organized list, a table, AND a tree diagram.

2. A chef’s salad at a local restaurant comes with a choice of French, ranch, or blue cheese dressings and optional toppings of cheese, turkey, and eggs. Draw a tree diagram to represent the sample space for salad orders.

3. New cars are available with a wide selection of options for the consumer. One option is chosen from each category shown. How many different cars could a consumer create in the chosen make and model? Homework – Page 902 – 904 ( 6 – 19, 22, 23) ALL Section 2 – Probability with Permutations and Combinations  I can use permutations with probability.  I can use combinations with probability.

 Permutation – an arrangement of objects in which order is important. The number of permutations of ‘n’ distinct objects taken ‘r’ at a time is denoted by:

 Factorial – of a positive integer ‘n’, written n!, is the product of the positive integers less than or equal to n. (n! = n (n-1) (n-2)……2-1, where 0! = 1)

 Permutations with Repetition – the number of distinguishable permutations of ‘n’ objects in which one object is repeated ‘r’ times, another is repeated ’r’ times and so on is:

 Circular Permutations – objects arranged in a circle or loop. The number of distinguishable permutations of ‘n’ objects arranged in a circle with no fixed reference point is:  Combinations – the number of combinations of ‘n’ distinct objects taken ‘r’ at a time is noted by:

1. Eli and Mia, along with 30 other people, sign up to audition for a talent show. Contestants are called at random to perform for the judges. What is the probability that Eli will be called to perform first and Mia will be called second?

2. There are 12 puppies for sale at a local pet shop. Four are brown, four are black, three are spotted, and one is white. What is the probability that all the brown puppies will be sold first?

3. A box of floor tiles contains 5 blue (bl) tiles, 2 gold (gd) tiles, and 2 green (gr) tiles in random order. The desired pattern is bl, gd, bl, gr, bl, gd, bl, gr, bl. If you selected a permutation of these tiles at random, what is the probability that they would be chosen in the correct sequence?

4. Find the indicated probability. Explain your reasoning. (a) If 8 students sit at random in a circle of chairs shown, what is the probability that the students sit in the arrangement shown? (b) You purchased a box of 8 crayons. If the crayons are packaged in random order, what is the probability that the crayon on the far left is red?

5. A set of alphabet magnets are placed in a bag. If 5 magnets are drawn from the bag at random, what is the probability that they will be the letters a,e,I,o, and u?

Homework – Page 911 – 913 (6 – 20)ALL Section 3 – Geometric Probability  I can find probability by using length.  I can find probability by using area.

 Geometric Probability – probability that involves a geometric measure such as the length or area.

1. Point Z is chosen at random on AD. Find the probability that Z is on AB.

2. Halley’s Comet orbits the earth every 76 years. What is the probability that Halley’s Comet will complete an orbit within the next decade?

3. The targets of a dartboard are formed by 3 concentric circles. If the diameter of the center circle is 4 inches and the circles are spread 3 inches apart, what is the probability that a player will throw a dart into the center circle? 4. Use the spinner to find each probability.

(a) P (Pointer landing on section 3) (b) P (Pointer landing on section 1)

Homework – Page 918 – 920 (6 – 22, 34) ALL Section 4 – Simulations  I can design simulations to estimate probabilities.  I can summarize data from simulations.

 Probability Model – A mathematical model used to match a random phenomenon.

 Simulation – the use of probability model to recreate a situation again and again so that the likelihood of various outcomes can be estimated. To do a simulation follow these steps:  Determine each possible outcome and its theoretical probability.  State any assumptions.  Describe an appropriate probability model for the situation.  Define what a trial is for the situation and state the number of trials to be conducted.

 Random Variable – a variable that can assume a set of values, each with fixed probabilities.

 Expected Value – a mathematical expectation – average value of a random variable that one “expects” after repeating an experiment or simulation a theoretically infinite number of times. Steps include:  Multiply the value of X by its probability of occurring.  Repeat Step 1 for all possible values of X.  Find the sum of the results.

1. Maria got a hit 40% of the time she was at bat last season. Design a simulation that can be used to estimate the probability that she will get a hit at her next at bat this season. 2. A survey of Longmeadow High School students found that 30% preferred cheese pizza, 30% preferred peppers and onions, 20% preferred sausage. Design a simulation that can be used to estimate the probability that a Longmeadow High School student prefers each of these choices.

3. Refer to the simulation in example 1 above. Conduct the simulation and report the results, using appropriate numerical and graphical summaries.

4. Suppose that an arrow is shot at a target. The radius of the center circle is 3 inches, and each successive circle has a radius 5 inches greater than that of the previous circle. The point value for each region is shown. (a) Let the random variable Y represent the point value assigned to the region on the target. Calculate the expected value E(Y) for each shot of the arrow.

(b) Design a simulation to estimate the average value, or the average of the results of your simulation, of shooting this game. How does this value compare with the expected value you found in part (a)?

Homework – Page 927 – 928 (5 – 17)ODD and 12 Section 5 – Probabilities of Independent and Dependent Events  I can find probabilities of independent and dependent events.  I can find probabilities of events given the occurrence of other events.

 Compound Event - or composite event – consists of two or more simple events. Example: Events A and B are: o Independent Events – if the probability that A occurs does not affect the probability that B occurs o Dependent Events – if the probability that A occurs in some way changes the probability that B occurs  Probability of TWO INDEPENDENT events – occur by the product of the probabilities of each event.  Probability of TWO DEPENDENT events – occur when the product of the probability of the first event occurs and the probability that the second event occurs after the first event has occurred.  Conditional Probability –

1. Determine whether the events are independent or dependent. Explain your reasoning. (a) A die is rolled, and then a second die is rolled.

(b) A card is selected from a deck of cards and not put back. Then a second card is selected.

2. Michelle and Christina are going out to lunch. They put 5 green slips of paper and 6 red slips of paper into a bag. If a person draws a green slip, they will order a hamburger. If they draw a red slip, they will order pizza. Suppose that Michelle draws a slip. Not liking the outcome, she puts it back and draws a second time. What is the probability that on each draw her slip is green?

3. Refer to the last example. Suppose that Michelle draws a slip and does not put it back. Then her friend Christina draws a slip. What is the probability that both friends draw a green slip?

4. Mr. Monroe is organizing the gym class into two teams for a game. The 20 students randomly draw cards numbered with consecutive integers from 1 to 20. -Students who draw odd numbers will on the Red team. -Students who draw even numbers will be on the Blue team. If Monica is on the Blue team, what is the probability that she drew the number 10?

(a) 1/20 (b) 1/10 (c) 9/20 (d) ½

Homework – Page 935 – 936 (6 – 18, 23)ALL Section 6 – Probabilities of Mutually Exclusive Events  I can find probabilities of events that are mutually exclusive and events that are not mutually exclusive.  I can find probabilities of complements.  Mutually Exclusive – if two events cannot happen at the same time. If two events have NO outcomes in common.  Probability of Mutually Exclusive Events – the probability that A or B occurs is the SUM of the probabilities of each individual event. P(A or B) = P(A) + P(B).  Probabilities of Events that are NOT Mutually Exclusive – the probability that A or B occurs is the SUM of their individual probabilities minus the probability that both A and B occur. P(A or B) = P(A) + P(B) = P(A and B)  Probability of the Complement of an Event – the probability that an event will not occur is equal to 1 minus the probability that the event will occur. For event A, P(not A) = 1 – P(A)  IMPORTANT – USE THE TABLE ON PAGE 941 TO HELP SOLVE THESE PROBLEMS!!!

1. Han draws one card from a standard deck. Determine whether the events are mutually exclusive or not mutually exclusive. Explain your reasoning. (a) an ace or a 9 (b) a king or a club

2. Trevor reaches into a can that contains 30 quarters, 25 dimes, 40 nickels and 15 pennies. What is the probability that the first coin he picks is a quarter or a penny?

3. Use the table from Example 3 in the book. What is the probability that Namiko selects a watercolor or a landscape?

4. Miquel bought 15 chances to pick one red marble from a container to win a gift certificate to the bookstore. If there is a total of 200 marbles in the container, what is the probability that Miquel will not win the gift certificate?

5. A survey of Kingston High School students found that 63% of the students had a cat or a dog for a pet. If two students are chosen at random from a group of 100 students, what is the probability that at least one of them does not have a cat or a dog for a pet?

Homework – Page 943 – 944 (8 – 28) ALL