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Chapter 16: Probability. Section 16.1: Basic Principles of Probability. Terminology. Chance process: experiment or situation where we know the possible outcomes, but not which will occur at a given time Sample space: the set of different possible outcomes
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Chapter 16: Probability Section 16.1: Basic Principles of Probability
Terminology • Chance process: experiment or situation where we know the possible outcomes, but not which will occur at a given time • Sample space: the set of different possible outcomes • Ex: For the chance process of flipping a coin, the sample space consists of the 2 outcomes of the coin landing on either heads or tails • Events: collections of outcomes • Ex: If you draw a card from a standard 52 card deck, one event is drawing a face card or a spade.
Definition of Probability • The theoretical probability of a given event is the fraction or percentage of times that the event should occur • Ex: The probability of flipping a coin and having it land on heads is 1/2 or 50%. • We use the shorthand notation P(heads)=1/2. • In general, P(event)=. • Probabilities are always between 0 and 1 (or 0% and 100%) • Ex: If drawing a single card, P(spade and heart)=0% and P(heart or diamond or spade or club)=100%.
Principles of Probability • If two events are equally likely, then their probabilities are equal. • The probability of an event is the sum of the probabilities of the distinct outcomes that compose that event. • If an experiment is performed many times, then the fraction of times it occurs should be similar to the probability. Misconception: The probability of an event occurring at least once within multiple experiments is not the sum of the probabilities for each experiment. Ex: The probability of having heads land at least once when flipping a coin two times is not
Example problem 2. The probability of an event is the sum of the probabilities of the distinct outcomes that compose that event. Ex 1: If you draw a card from a standard 52 card deck, find the following probabilities: P(queen or jack), P(queen or a spade).
Uniform Probability Models • Uniform Probability Model:chance process with all distinct possible outcomes being equally likely • N possible outcomes ⇒ 1/N probability for each outcome • Ex: Rolling a 6 sided die: P(roll a 1)=P(roll a 2)=P(roll a 3)=P(roll a 4)=P(roll a 5)=P(roll a 6)=1/6 • Ex 2: What is the probability of rolling an even number if you roll a 6 sided die?
Experimental Probability • Experimental or empirical probability: the fraction of times an event occurs after performing the event a number of times • Ex: When flipping a coin 20 times, if it lands on heads 9 times, the experimental probability of getting heads is 9/20=45%. • See Activity 16E
Multistage Experiments • Multistage Experiment: consists of performing several experiments in a row • The events in the sample space of multistage experiments are called compound events. • Ex 3: The board game Twister involves two spinners: one which selects a body part (left hand, right hand, left foot, or right foot) and one that selects a color to place that body part (red, green, blue, or yellow). How many outcomes are there for each turn in which you spin both of the spinners?
A Different Type of Multistage Experiment • Multistage experiments with dependent outcomes are ones in which one stage affects the upcoming stages. • Ex 4: If you are dealt two cards from a 52 card deck, what is the probability that you are dealt 2 aces?
Section 16.3: Calculating Probabilities in Multistage Experiments
Independent Vs Dependent Outcomes • Def: Outcomes of a multistage experiment are independent if the probability of each stage is not influenced by the previous stage’s outcome • Ex’s: Flipping a coin multiple times Drawing cards with replacement • Def: Outcomes are dependent if the probability of each stage is affected by the outcome of the previous stage • Ex’s: Drawing/dealing cards without replacement The probability a baseball player gets a hit
Example Problem • Ex 1: If you flip a coin 4 times, what is the probability that it lands on tails at least 3 times?
Another Example • Ex 2: You have a bag with 3 red marbles and 1 blue marble. If you reach in and randomly grab 2 marbles, what is the probability of picking the blue marble? • See Activity 16H for more examples
Expected Value • Def: For an experiment that has numerical outcomes, the expected value of the experiment is the average outcome of the experiment over the long term. • Ex 3: If the Kentucky Lottery sells a scratch off ticket for $2 that has a 1% chance of winning $100 and a 10% chance of winning $5, how much money does the state expect to make off each ticket sale?
Section 16.4: Calculating Probability with Fraction Multiplication
Using Fraction Multiplication • Ex 4:Use fraction multiplication to find the probability of rolling a 12 when you roll two 6-sided dice. • See Activity 16L for another example.
The Monty Hall Problem • You are on a game show and will win the prize behind your choice of one of 3 doors. Behind one door is a brand new car! Behind the other two doors are goats. You pick a door, say Door # 1, and the host, who knows what’s behind each door, opens another door, say Door #3, which has a goat behind it. The host then asks you, “Do you want to switch your choice to Door #2?” Should you make the switch?
