T HEORETICAL P ROBABILITY Lesson 16

WARM UP Name the property illustrated = = 0 2(x + 5) = 2x (3 + 5) = (2 + 3) + 5

WARM UP- SOLUTIONS Name the property illustrated = 3 Identity = 0 Inverse 2(x + 5) = 2x + 10 Distributive Property 2 + (3 + 5) = (2 + 3) + 5 Associative Property

W HAT IS P ROBABILITY ? The probability of an outcome is a ratio of It is the likelihood of an event happening. Examples The probability of rolling a 2 on a die is 1/6 The probability of a heads is ½ The probability of drawing a king from a standard deck is 4/52 = 1/13 number of ways the desired outcome can occur the total number of possible outcomes.

EXAMPLE 1 Kylee wrote the names of the months of the year on slips of paper and put them in a box. If she picks out one slip, find the following probabilities. P(picking a month that has exactly 4 letters) P(picking a month that begins with a vowel) P(picking a month that has at least 3 letters)

EXAMPLE 1- SOLUTIONS P(picking a month that has exactly 4 letters) There are 2 months that have 4 letters: June, July 2/12 = 1/6 P(picking a month that begins with a vowel) There are that 3 months that start with a vowel: April, August, October 3/12 = 1/4 P(picking a month that has at least 3 letters) All of the months have at least 3 letters 12/12 = 1 January February March April May June July August September October November December

EXAMPLE 2 Patrick has a bag of m&ms. It contains 8 blue, 15 green, 3 orange and 12 red. Patrick will pick a candy out of the bag without looking. Find each probability. P(green) P(brown) P(orange or green)

EXAMPLE 2- SOLUTIONS Patrick has a bag of m&ms. It contains 8 blue, 15 green, 3 orange and 12 red. Patrick will pick a candy out of the bag without looking. Find each probability. P(green) There are 38 total candies, 15 are green. P(green) = 15/38 P(brown) There are no brown candies. P(brown) = 0/38 = 0 P(orange or green) = 18 18/38 = 9/19

EXAMPLE 3 Patrick has a bag of m&ms. It contains 8 blue, 15 green, 3 orange and 12 red. Patrick will pick a candy out of the bag without looking. Which color is he most likely to pick from the bag? Which color is he least likely to pick from the bag?

EXAMPLE 3- SOLUTIONS Patrick has a bag of m&ms. It contains 8 blue, 15 green, 3 orange and 12 red. Patrick will pick a candy out of the bag without looking. Which color is he most likely to pick from the bag? Since there are more green than any other color, green is the most likely color to occur. Which color is he least likely to pick from the bag? Since there are only 3 orange, and more of every other color, orange is least likely to occur.

I NDEPENDENT VS D EPENDENT E VENTS Independent events do not effect each other. Events “with replacement”, such as drawing a card and placing it back in the deck. Rolling dice (a 3 on the first roll has no effect on the second roll). Dependent events effect each other. Events “without replacement”, such as dealing cards. Choosing students to be on a team.

COMBINED PROBABILITIES When you want to find the probability of more than one event at a time, multiply each individual probability together. Examples: Find the probability of rolling a 3 followed by a 2. Each of these events are independent. The probability of rolling a 3 is 1/6, the probability of tolling a 2 is 1/6 so the probability of one followed by the next is: (1/6)(1/6)= 1/36 Find the probability of drawing a king followed by an ace. These events are dependent. The probability of drawing a king is 4/52. Once the king is drawn there are only 51 cards left, so the probability of an ace is 4/51. The probability of both is (4/52)(4/51) = 16/2652 = 4/663

E XAMPLE 4 Determine if the following events are independent or dependent. Then find the probability There are 6 questions on a multiple choice test. Each question has 4 answer choices. What is the probability of a person guessing all of them correctly? There are 6 questions on a matching test with exactly one answer paired with each question. What is the probability that a person who guesses on all five questions will answer them all correctly?

E XAMPLE 4- S OLUTIONS Determine if the following events are independent or dependent. Then find the probability There are 6 questions on a multiple choice test. Each question has 4 answer choices. What is the probability of a person guessing all of them correctly? Each guess is independent. Getting one right does not effect the chances of getting the next one right.

E XAMPLE 4- S OLUTIONS Determine if the following events are independent or dependent. Then find the probability There are 6 questions on a matching test with exactly one answer paired with each question. What is the probability that a person who guesses on all five questions will answer them all correctly? These events are not independent. Once you get the first question right there are only 5 more to choose from.

E XAMPLE 5 There are 25 students in a class. 12 of them are girls and 13 are boys. If all of their names are placed in a hat, what is the probability that a girl will be drawn first followed by a boy? What is the probability that 2 boys names will be drawn?

E XAMPLE 5- S OLUTION There are 25 students in a class. 12 of them are girls and 13 are boys. If all of their names are placed in a hat, what is the probability that a girl will be drawn first followed by a boy? These are dependent events. What is the probability that 2 boys names will be drawn? These are dependent events.

E XAMPLE 6 The spinner is spun once. What is the probability that the number spun is divisible by 3? What is the probability that the number spun is less that 5? What is the probability that the number spun is even or a 5?

E XAMPLE 6- SOLUTIONS The spinner is spun once. What is the probability that the number spun is divisible by 3? 3 and 6 are divisible by 3 2/8 = 1/4 What is the probability that the number spun is less that 5? 1,2,3,4 are less than 5 4/8 = 1/2 What is the probability that the number spun is even or a 5? 2,4,6,8,5 5/8 There are 8 values on the spinner, each one is the same size so they each have the same probability of occurring: 1/8

F UNDAMENTAL C OUNTING P RINCIPLE A pizza shop wants to see how many outcomes can be made with the following options. Crust: pan, thin. Cheese: mozzarella, Parmesan; Toppings: Pepper, Ham, Sausage How many different pizzas can be made choosing 1 crust, 1 cheese and 1 topping. 2 crust x 2 cheese x 3 toppings = 12 possible outcomes Tree diagrams can be used to show the outcome set of a situation.

EXAMPLE 7 The first 3 questions on a history test are true/false. Make a tree diagram to show how many different ways the 3 questions can be answered. (use T for true and F for false )

EXAMPLE 7- SOLUTION The first 3 questions on a history test are true/false. Make a tree diagram to show how many different ways the 3 questions can be answered. True False TrueFalse True False TrueFalse **Notice there are 2 x 2 x 2 = 8 ways that the 3 questions can be answered

EXAMPLE 8 A coin is tossed up in the air 4 times. Makes a tree diagram to show how many different ways all 4 tosses can land. (Use H for heads and T for tails.)

Heads Tails HeadsTails Heads Tails HeadsTails Heads Tails HeadsTails Heads Tails HeadsTails EXAMPLE 8- SOLUTION A coin is tossed up in the air 4 times. Makes a tree diagram to show how many different ways all 4 tosses can land.

EXAMPLE 9 Using the digits 0-9, how many different 4 digit numbers are possible? (Repetition of digits is allowed.)

EXAMPLE 9- SOLUTION Using the digits 0-9, how many different 4 digit numbers are possible? (Repetition of digits is allowed.) (10)(10)(10)(10) = 10000

EXAMPLE 10 Using the digits 0-9, how many different 4 digit numbers are possible if the first digit cannot be zero? (Repetition of digits is allowed.)

EXAMPLE 10- SOLUTION Using the digits 0-9, how many different 4 digit numbers are possible if the first digit cannot be zero? (Repetition of digits is allowed.) (9)(10)(10)(10) = 9000

EXAMPLE 11 The Math Club has 12 girls, 8 boys, and 4 adults chaperones going on a field trip. How many different groups of 1 girl, 1 boy, and 1 adult are there?

EXAMPLE 11- SOLUTION The Math Club has 12 girls, 8 boys, and 4 adults chaperones going on a field trip. How many different groups of 1 girl, 1 boy, and 1 adult are there? (12)(8)(4) = 384

EXAMPLE 12 Arizona license plates consist of 3 digits followed by 3 letters. How many different license are possible? (Repetition of digits and letters are allowed.) Set up, but do not multiply out.

EXAMPLE 12- SOLUTION Arizona license plates consist of 3 digits followed by 3 letters. How many different license are possible? (Repetition of digits and letters are allowed.) Set up, but do not multiply out. (10)(10)(10)(26)(26)(26)

EXAMPLE 13 A particular shirt comes in 2 colors, 2 styles, and 4 sizes. The following table shows all the choices. How many different shirts are possible? ColorStyleSize BlackCrew neckSmall GoldV-neckMedium Large X-Large

EXAMPLE 13- SOLUTION A particular shirt comes in 2 colors, 2 styles, and 4 sizes. The following table shows all the choices. How many different shirts are possible? (2)(2)(4) = 16 different shirts ColorStyleSize BlackCrew neckSmall GoldV-neckMedium Large X-Large

EXAMPLE 14 A meal at a certain restaurant includes one type of meat, potato, and vegetable. The following table show all the choices. How many meals are possible? MainSideVegetable BeefBaked PotatoPeas FishScallopedCarrots French Fries

EXAMPLE 14- SOLUTION A meal at a certain restaurant includes one type of meat, potato, and vegetable. The following table show all the choices. How many meals are possible? (2)(3)(2) = 12 different meals MainSideVegetable BeefBaked PotatoPeas FishScallopedCarrots French Fries