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Learn fundamental concepts in probability theory, from experiments and sample spaces to event probabilities and vocabulary. Practice with coin flips, dice rolling, cards, and more.
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Probability Basics • Experiment • 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 —General Definition where P(E) represents the probability of an event E occurring, n(E) represents the number of individual outcomes in the event E, and n(S) represents the number of individual outcomes in the sample space S.
Flip a coin • A well-known statistician named Karl Pearson once flipped a coin 24,000 times and recorded ________ “heads”; this result is extremely close to the theoretical expected value! • P(H) = _____ P(T) = _____ • Expected # of H = P(H) x 24,000 = _____
A A B C D Spinners Spin each spinner once. Find the probability that the spinner lands in region A. B A C
Spinners If S = {1, 2, 3, 4, 5, . . . , 22, 23, 24}, find the probability of the spinning 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
Rolling Dice Roll a single die once. Find the following probabilities: • P(number greater than 4 or less than 2) • P(odd or even number) • P(number greater than 10) • P(at least 3)
Probability Vocabulary • Complementary event • Everything else (besides the outcomes in the event) in the sample space • Examples: • If A = “roll a 1 or a 2 on a die”, then “A complement” = “roll a 3, 4, 5, or 6 on a die”. • If R = “it rains today”, then R complement = “it doesn’t rain today”.
Standard 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 events (Disjoint sets): • When one event occurs, the other cannot possibly occur; the events have no overlap • Example: • If A = “roll an even number” and B = “roll a 3 or a 5”, find P(A or B) and find P(A and B).
Probability of A or B • Mutually exclusive events • Non-mutually exclusive events
Probability of A or B • Draw a card out of a standard 52-card deck. Find the probability that the card is either: (a) a black card or an ace (b) a red card or a club • Roll a die once. If A = “roll an even number” and B = “roll a 5 or a 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. A tree diagram helps!)
Fundamental Counting Principle • How many outcomes are there for flipping 3 coins? • How many outcomes are there for rolling 2 dice? • If I have 6 pairs of pants and 8 shirts from which to choose, how many outfits can I pick?
Fundamental Counting Principle • If automobile license plates consist of 4 letters followed by 3 digits (and repetition of letters and digits is allowed), how many different license plates are possible? (This time, a tree diagram isn’t encouraged.)
Multi-stage Experiments • For any multi-stage experiment, the probability of the outcome along any path of the tree diagram is equal to the product of the probabilities along the path.
Toss 2 coins • List the sample space. Use set notation and a tree diagram. • Find the probability of at least one head.
The Problem • If the chance for success on the first stage of a rocket firing procedure is 96%, the chance for success on the second stage is 98%, and the chance for success on the final stage is 99%, find the probability for success on all 3 stages of the rocket firing procedure.
Rolling Two Dice • Find the probability of a 3 on the first roll and a 3 on the second roll of a die. • 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.
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 2nd 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).
Draw a ball from a container, don’t replace it, and then draw a 2nd ball. (dependent events) • Find P(red, then green). • Find P(no ball is red). • Find P(same color ball).
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
2 2 1 1 3 4 Geometric Probabilities • If a dart hits the target below, find the probability that it hits somewhere in region 1. The radius of the inner circle is 1 unit, and the radius of the outer circle is 2 units.
For a challenge, or two, or three! • “Pascal’s Probabilities” • “The Prisoner Problem” • “The Birthday Problem”
Using Simulations • Flipping a coin • Rolling a die • Find the probability of a married couple having 2 boys and 2 girls.
Odds • Find the odds for tossing a “head” on a fair coin. • Find the odds for rolling a sum of 7 on the roll of two dice. • Find the odds for drawing a card valued from 1 (ace) to 8, inclusive, from a standard 52-card deck.
Conditional Probabilities • When the sample space of an experiment is affected by additional information
Conditional Probabilities • If A = “getting a tail on the 1st toss of a coin” and B = “getting a tail on all three tosses of a coin”, find P(B|A). • What is the probability of rolling a 6 on a fair die if you know that you rolled an even number?
Expected Value • If, in an experiment, the possible outcomes are numbers a1,a2,a3, . . . , an occurring with probabilities p1,p2,p3, . . . , pn, respectively, then the expected value, E, is given by the equation E = a1 p1 + a2 p2 +a3 p3 + . . . , + an pn.
Expected Value (level 1) • Flip a coin 1,000 times. How many heads do you expect? • Roll a pair of dice 60 times. How many times do you expect a sum of 5?
A A B C D Expected Value (level 2) • If a player gets $2 if the spinner lands on A, $4 for landing on B, $4 for C, and $1 for D, what is the expected payoff for this game? • If the game costs $3 to play, is this a fair game?
Factorial Notation • 0! = 1 (by definition) • 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 given by
Permutations (Correspondences) • How many different ways can 4 swimmers (Al, Betty, Carol, 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 in the field and at bat?
Combinations • From n objects, choose subsets of size r (order is unimportant). The number of ways this can be done is given by
Combinations • With 9 club members, how many different committees of 4 can be selected to attend a conference? • Braille Activity
Permutations & Combinations • How many games are played in a women’s soccer conference if there are 8 teams and all teams play one another once? • 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 (withpermutations & 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?
