Transcript Chapter 16: 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,
# of outcomes for that event
P(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
1. If two events are equally likely, then their probabilities are equal.
2. The probability of an event is the sum of the probabilities of the distinct
outcomes that compose that event.
3. 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
1
2
1
2
coin two times is not + = 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?