Section 16.1: Basic Principles of Probability 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)= # of outcomes for that event size of sample space . 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 1 2 + 1 2 =1.

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

Section 16.2: Counting the Number of Outcomes

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?