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Unit: Probability
Assignments:
Due DateClasswork (in groups)Homework Assignments (individually)Activity 1:
Notes & Examples
Exploration (1 3)
Discussion (1 4)6-9 Practice (1 21)
6-9 Enrichment (1 11)
13-3 Practice (1 9)Activity 2:
Notes & Examples
Exploration (1 9)
Discussion (1 5)Assignment #1:
13-5 Practice (1 8)
13-6 Practice (1 10)Assignment #2:
12-9 Skills Practice (1 22)
13-6 Enrichment (1 6)Activity 3:
Notes & Examples
Exploration (1 8)
Discussion (1 4)Handout (1 2)Activity 4:
Notes & Examples
Exploration (1 8)Handout (1 9)Mini-ProjectHandout due ( / )Unit TestTest Review due ( / )
Prerequisite Knowledge:
Calculate combinations and permutations
Activity 1
Objectives:
Define basic probability vocabulary
Calculate simple probability
Vocabulary:
Experiment an activity that can be repeated and leads to observable outcomes
Event the outcomes of an experiment
Sample Space the set of all possible outcomes of an experiment
Probability the likelihood that a specific event(s) will occur
Concepts:
To calculate simple probability use the formula:
EMBED Equation.DSMT4
In some cases (like spinners and dart boards), probability is based on the fraction of the whole area that the desired outcome represents. (See below for example.)
Probability is usually expressed as a fraction, but it can also be expressed as a decimal or a percentage. Probability is always a number between 0 and 1 (or 0% to 100%).
Examples:
What is the sample space flipping one coin?
What is the sample space for flipping two coins?
What is the probability of flipping a head on a fair coin?
What is the probability of flipping two heads on two fair coins?
Assuming you hit the target, what is the probability of hitting the bulls eye? (Hint: Youll need to find the area of both circles first.)
SHAPE \* MERGEFORMAT
Activity 2
Objectives:
Use Venn diagrams to calculate probability
Differentiate between independent and dependent events
Calculate probabilities for compound events
Vocabulary:
Independent Events events whose probabilities have no effect on each other
Dependent Events events whose probabilities have an effect on each other
Mutually Exclusive Events events that cannot happen at the same time
Complementary Events (symbolized with ~ or ) events that are negations of each other (e.g. A and not A); the sum of the probabilities of complementary events is 1
Conditional Probability (symbolized P(B|A)) probability that one event happens given that the other event has already happened
Concepts:
Venn diagrams and tree diagrams are useful visual aids for determining probabilities. Use them whenever possible.
There are several types of Venn diagrams; the two common types used in probability are pictured below. The first type represents the situation when two events can happen separately or together. The second type represents two events that cant happen together (mutually exclusive events).
SHAPE \* MERGEFORMAT SHAPE \* MERGEFORMAT Many people often confuse independent events with mutually exclusive events. Complementary events are mutually exclusive of each other. Independent events are not mutually exclusive of each other.
The probability of one event OR the other can be calculated by adding the probabilities of the two events and then subtracting the probability that they happen together (if they can happen together).
P(A EMBED Equation.3 B) = P(A) + P(B) P(A EMBED Equation.3 B) (if A and B are not mutually exclusive)
P(A EMBED Equation.3 B) = P(A) + P(B) (if A and B are mutually exclusive)
The probability of one event AND the other can be calculated by multiplying the probabilities of the two events together. (If the events are dependent events, youll have to multiply the probability of the first event happening by the conditional probability for the second event happening.)
P(A EMBED Equation.3 B) = P(A) " P(B|A) (if A and B are dependent events)
P(A EMBED Equation.3 B) = P(A) " P(B) (if A and B are independent events)
Examples:
Draw a Venn diagram for three mutually exclusive events and a Venn diagram for three non-mutually exclusive events. Using colors, label the sections.
In a given population, the probability of having brown hair is 55%, the probability of having brown eyes is 70%, and the probability of having both is 40%. Use a Venn diagram to find the following:
The probability of having brown hair but not brown eyes
The probability of having brown eyes but not brown hair
The probability of having neither brown hair nor brown eyes
Tell whether the following events are independent or dependent.
Flipping a coin and then flipping it again
Pulling a marble out of a bag and then pulling out another marble without replacing the first
Pulling a marble out of a bag and then pulling out another marble after replacing the first
A slot machine has three shapes (triangle, square, and circle) on each of three reels. Find the following:
A tree diagram illustrating all possible outcomes with the correct probabilities for each branch
The probability of getting a circle on the first reel
The probability of getting a circle on the first, second, and third reel
The probability of getting a circle on the first or second reel
There 6 blue marbles and 4 red marbles in a bag. Marbles are drawn one at a time without replacement. Find the following:
The probability of getting a red marble on the second draw given that a blue marble was drawn first
The probability of getting a blue marble on the first draw and a red marble on the second draw
The probability of getting a red and a blue marble
The probability of getting a red or a blue marble.
Activity 3
Objectives:
Compare and contrast experimental and theoretical probability
Use simulations to estimate theoretical probability
Vocabulary:
Theoretical Probability the probability that an event should occur
Experimental Probability the actual probability of an event occurring
Concepts:
When the theoretical probability is unknown or impossible to calculate, repeated trials with the experimental probability can often be used to estimate the theoretical probability. The more trials performed, the closer to the theoretical probability the experimental probability comes. (This is known as the Law of Large Numbers.) This theory is the idea behind polls and surveys. The pollsters must ask many people their opinions on a topic before it is accepted as an accurate probability for the entire population.
Examples:
What is the theoretical probability of getting heads when a fair coin is flipped? If you flipped a coin 10 times, how many times would you expect to get heads? If you actually performed this experiment, do you think youd ALWAYS get that many heads on 10 flips? Why or why not?
A coin was flipped 30 times. The results of these trials were recorded in the table below.
Outcome
Frequency
Heads
11
Tails
19
What is the experimental probability of getting heads?
What is the experimental probability of getting tails?
Do you think this is a fair coin? Why or why not?
Activity 4
Objectives:
Calculate expected value
Vocabulary:
Odds the ratio of favorable outcomes to unfavorable outcomes
Fair Game a gambling game in which the expected value equals the cost to play
House/Player Advantage in an unfair gambling game, the side with the expected value in their favor
Expected Value the predicted value of the outcome of a probability experiment
Concepts:
The expected value can be calculated by using the following formula:
EMBED Equation.3
To calculate expected value:
List the entire sample space (including losing outcomes)
Find all the probabilities of all the events in the sample space
Multiply each of the probabilities with their corresponding values
Add up all of your answers from step #3
Examples:
Given the probability, what are the odds?
3 to 5
5 to 345%A game is unfair when the cost to play does not equal the expected value. What two situations could make this happen?
In Probability & Statistics, homework is worth 25%, tests are worth 30%, classwork is worth 25%, projects are worth 10%, and participation is worth 10%. If Albert got an 85 on his homework, 75 on his tests, 95 on his project, 90 on his classwork, and 80 for participation, what is his final grade?
Homework =
Classwork =
Project =
Tests =
Participation =
__________________________
A mysterious card-playing squirrel offers you the chance to join in his game. To join, though, you have to pay him $2. If you pick a spade from the deck, you win $9. What is the expected value of this game, and is this a fair game? If so, explain why. If not, who has the advantage?
Three dogs (Spot, Rover, and Fido) have raced against each other many times. Spot wins about 45% of the time; Rover wins about 35% of the time; Fido wins 20% of the time. Find the following:
The odds for each dog
The expected value for each dog if you bet $50
A roulette wheel has 38 spaces, labeled 00 to 36. Two spaces are green (0 and 00); 18 are red; 18 are black. You can place your bet on a number (1 36) or a color (red or black only). For every dollar you place on the correct color, you get a dollar back. If you do not choose the correct color, you lose your bet. Find the following:
If you bet $1 on black, what is the expected value?
If you place $30 on red, what is the expected value?
A car insurance company charges an annual premium of $1500 for a policy that will cover bodily injury liability and property damage liability. The benefits are as follows:
Bodily Injury up to $50,000
Property Damage up to $10,000
According to the companys statistics, the probability of damaging property in any one year is .08, the probability of causing an injury is .005, and the probability of damaging property and causing an injury is .003. Find the following:
The probability that the policyholder damages property but does not cause an injury
The probability that the policyholder does not damage property but causes an injury
The probability that the policyholder does not damage property or cause injury
The companys expected annual profit per policyholder
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