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Probability Basics Experiment Sample space Event an activity with observable results or outcomes Sample space the set of all possible outcomes for an experiment Event any subset of the sample space

Probability Basics where P(E) means the probability of an event occurring, n(E) means the number of individual outcomes in the event, and n(S) means the number of individual outcomes in the sample space.

Flip a coin A well-known statistician named Karl Pearson once flipped a coin 24,000 times and recorded _______ “heads”; this result was extremely close to the theoretical probability and expected number of heads. P(H) = _____ E(H) = _________

Spinners A B C A B C D Spin each spinner once. Find P(A).

Spinners If S = {1, 2, 3, 4, 5, . . ., 22, 23, 24}, find the probability of a: Prime number Even number Number less than 10 Number less than 3 or greater than 17 Number less than 12 and greater than 9

Dice Roll a single die once. Find the following probabilities: P(number greater than 4 or less than 2) P(odd or even) P(greater than 10) P(at least 3)

Experimental Probabilities Conduct a poll to determine the probability: That an ABAC transfer student drives a truck. That an ABAC student has enjoyed a meal at the Donaldson Dining Hall. That an ABAC student has visited the BCM.

More vocabulary Complementary events Everything else in the sample space Examples: If A = rolling a 1 or a 2 on a die, “A complement” is rolling a 3, 4, 5, or 6 If R = it rains today, = it doesn’t rain today

Cards Find the probability of drawing an ace from a standard deck of playing cards. Find P(face card) Find P(card with a value between 4 and 9)

More vocabulary Mutually exclusive (disjoint) events When one event occurs, the other cannot possibly occur Examples: If A = even # on the roll of a die and B = roll a 3 or a 5,

P(A or B) Mutually exclusive events Non-mutually exclusive events

P(A or B) Draw a card out of a standard 52-card deck. Find the probability that the card is either black or an ace. Roll a die once. If A = “even number on the die” and B = “rolling a 5 or 6”, find P(A or B).

Fundamental Counting Principle If event M can occur in m ways and after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m x n ways. P.S. This rule extends to more than two events.

Fundamental Counting Principle How many outcomes are there for flipping 3 coins? How many outcomes are there for rolling 2 dice? 3 dice? If an ice cream shop has 32 flavors from which to choose and 7 toppings, how many different possibilities can I choose?

Fundamental Counting Principle If automobile license plates consist of 4 letters followed by 3 digits, how many different license plates are possible if letters and digits may be repeated?

Multi-Stage Experiments For any multi-stage experiment, the probability of the outcome along any path of a tree diagram is equal to the product of the probabilities along the path.

Problem If the chance for success on the first stage of a rocket firing procedure is 96%, the second stage is 98%, and the final stage is 99%, find the probability of success on all three stages of the rocket firing procedure.

Flipping coins List the sample space for two coins. (Use a tree diagram.) Find the probability of at least one head. List the sample space for three coins. (Use a tree diagram.) Find the probability of exactly two heads.

Rolling Two Dice List the sample space. Find the probability of a 3 on the first and a 3 on the second. Find the probability of a sum of 7. Find the probability of a sum of 10 or more. Find the probability that both numbers are even.

Rolling Two Dice

Independent events When the outcome of one event has no influence on the outcome of a second event, the events are independent. For any independent events A and B, P(A and B) = P(A) x P(B).

Draw a ball from a container, replace it, and then draw a second ball. Find the probability of a red, then a red. Find P(no ball is red). Find P(at least one red). Find P(same color).

Dependent events When the outcome of one event has an influence on the outcome of a second event, the events are dependent.

Draw a ball from a container, don’t replace it, and then draw a second ball. Find the probability of a red, then a blue. Find P(no ball is red). Find P(same color).

A bag contains the letters of the word “probability”. Draw 4 letters, one by one, from the bag. Find the probability of picking the letters of the word “baby” if the letters are drawn With replacement Without replacement

Consider the following 2 containers. If a container above is selected at random, and then a letter is selected at random from the chosen container, what is the probability that the letter chosen is an A? MATH FAN #1 #2

For a challenge! The Prisoner Problem The Birthday Problem

Geometric Probabilities If a dart hits the target below, find the probability that it hits somewhere in region 1. 2 1 1 2 3 4 The radius of the inner circle is 2 units and the radius of the outer circle is 4 units.

Games A game is played by drawing 3 cards and replacing the card each time. A player wins if at least one face card is drawn. Find the probability of winning this game. Find the probability of a “Yahtzee” in one throw of 5 dice.

Using Simulations Flipping a coin Rolling a die Find the probability of a married couple having 2 baby girls.

Conditional Probabilities When the sample space of an experiment is affected by additional info

Conditional Probabilities If A = “getting a tail on the 1st toss of a coin” and B = “getting a tail on all 3 tosses of a coin”, find P(B|A). What is the probability of rolling a 6 on a fair die if you know you rolled an even number?

Conditional Probabilities With an auto insurance company, 60% of its customers are considered low-risk, 30% are medium-risk, and 10% are high-risk. After a study, the company finds that during a 1-yr period, 1% of the low-risk drivers had an accident, 5% of the medium risk drivers had an accident, and 9% of the high risk drivers had an accident. If a driver is selected at random, find the probability that the driver will have had an accident during the year.

Factorial Notation Compute:

Permutations From n objects, choose r of them and arrange them in a definite order. The number of ways this can be done is

Correspondences How many different ways can 4 swimmers (Al, Betty, Carole, and Dan) be placed in 4 lanes for a swim meet?

Permutations If there are 12 players on a little league baseball team, how many ways can the coach arrange batting orders, with 9 positions on the field and at bat?

Combinations From n objects, choose subsets of size r (order unimportant). The number of ways this can be done is

Combinations With 9 club members, how many different committees of 4 can be selected to attend a conference? Braille Activity

Permutations & Combinations There are 10 players on a U-6 soccer team, and the coach picks 5 starters. How many different groups of starters can the coach choose? There are 10 members of a club. How many different “slates” could the membership elect as president, vice-president, and secretary/ treasurer (3 offices)?

Probability (with permutations/combinations) Given a class of 12 girls and 9 boys, in how many ways can a committee of 5 be chosen? in how many ways can a committee of 3 girls and 2 boys be chosen? What is the probability that a committee of 5, chosen at random, consists of 3 girls and 2 boys?