Stats 231 Ch. 3 Basic Concepts of Probability and Counting
Find the probabilities (of a deck of cards): Event A: Drawing an Ace = __ Event B: Drawing a King = __ Event C: Drawing a Club = __ Event D: Drawing red cards = __ What is the probability of A or B? A or C?
-Ace: 4/52; King: 4/52; Club: 13/52; red card: 26/52 -P(A or B) = 4/52 + 4/52 = 0.1538 M.E. -P(A or C) = 4/52 + 13/52 - 1/52 = 0.3077 NOT M.E. **There are no unusual values.
An event that occurs with a probability of __ or less is typically considered unusual.
0.05
The probability of an impossible event is __; whereas, the probability of an event that is certain to occur is __.
0; 1
Types of Probability
1. Classical Approach 2. Empirical (or Statistical) Probability 3. Subjective Probability
Your band has written 11 songs and plans to record 5 of them for a CD. In how many ways can you arrange the songs on the CD?
11P5 = (11!)/(11-5)! = 55,440
From a group of 40 people, a jury of 12 people is selected. In how many ways can a jury of 12 people be selected?
40C12 = (40!)/(12!(40-12)!) = 5,586,853,480
What is the Example of Event and Sample Space for each Probability Experiment? A. Single birth B. Rolling a die C. Flipping a coin three times
A. EoE: having one girl; SS: {boy, girl} B. EoE: roll a 5; SS: {1 2 3 4 5 6} C. EoE: flipping 2 heads & a tails; SS: *use a tree diagram for all possible outcomes*
With the first total of survived: 203 and died: 122 (total: 325) and the total being 2201, what is the probability of a passenger on the Titanic surviving given they were in first class?
Event A: 1st Class Event B: survived P (survived/1st Class) = # of survivors/total 1st Class = 203/325 = 0.625.
Determine whether the events are mutually exclusive. Event A: Randomly selecting a registered Democrat. Event B: Randomly selecting a registered Republican.
Mutually Exclusive: cannot be registered as both Democrat and Republican.
Would getting 2 heads and 1 tails considered a simple event?
No because it has more than one outcome.
Determine whether the events are mutually exclusive. Event A: Randomly selecting someone taking a Stats course. Event B: Randomly selecting a female.
Not Mutually Exclusive: can have a female taking Stats.
A company supplies Blu-rays in lots of 50, and they have a reported defect rate of 0.5% so the probability of a disk being defective is 0.005. It follows that the probability of a disk being good is 0.995. What is the probability of getting atleast one defective disk in a lot of 50?
P(A*) = (0.995)^50 = 0.778 P(A) = 1 - P(A*) = 1 - 0.778 = 0.222 UNUSUAL: P(A) < 0.05? Not unusual.
A gender selection technique is designed to increase the likelihood that a baby will be a girl. Assume a preliminary test showed 20 couples gave birth to 20 babies and all were girls. If we assume the method has no effect, what is the probability of getting 20 girls by random chance?
P(girl) = 1/2 P(20 girls) = (1/2)^20 = 0.000000954 UNUSUAL: P(20 girls) < 0.05? Yes, it is unusual.
What is the probability of selecting a red marble from a bag with a total of 10 marbles and 3 of those marbles being red, not replacing it, and selecting another red marble?
P(red without replacing) = (3/10)(2/9) = 0.07 DEPENDENT
What is the probability of selecting a red marble from a bag with a total of 10 marbles and 3 of those marbles being red, replacing it, and selecting another red marble?
P(red) = (3/10)(3/10) = 0.09 INDEPENDENT
If we randomly select an adult in the U.S., the probability of selecting a smoker is 0.200. Find the probability of randomly selecting an adult in the U.S. and getting someone who does NOT smoke.
P(smoker) = 0.200 P(non-smoker) = 1 - 0.200 = 0.800 *the word "non-smoker" has a line over it representing the compliment (or opposite) of the event*
A recent survey of 1010 adults in the U.S. showed 202 of them smoke. Find the probability that a randomly selected adult in the U.S. is a smoker.
P(smoker) = number of smokers/total survey = 202/1010 = 0.2 P(non-smoker) = 1 - 0.2 = 0.8
Suppose we have the data on the outcomes of 44 successful surgeries and 6 unsuccessful surgeries our of a total of 50 surgeries. If 2 of the 50 subjects are randomly selected WITHOUT REPLACEMENT, find the probability that the first person had a successful surgery and the second person had an unsuccessful surgery.
P(success) = 44/50 P(unsuccessful) = 6/49 (44/50)(6/49) = 0.1078
Suppose we have the data on the outcomes of 44 successful surgeries and 6 unsuccessful surgeries our of a total of 50 surgeries. If 2 of the 50 subjects are randomly selected WITH REPLACEMENT, find the probability that the first person had a successful surgery and the second person had an unsuccessful surgery.
P(success) = 44/50 P(unsuccessful) = 6/50 (44/50)(6/50) = 0.1056
A student is to roll a die and flip a coin. How many possible outcomes will there be?
Possible Outcomes: (6)(2) = 12.
If boys and girls are equally likely, find the probability of getting three children of all the same gender.
Probability (2 same gender): 2/8 = 1/4 OR 0.25
Suppose you want to determine the probability of tossing a head with a fair coin. You toss a coin 10 times and get the following results of 7 heads and 3 tails. What is your empirical probability?
Total Toss = 10 Heads = 7 Probability = 7/10 It does not represent theoretical probability because it was not tossed enough times.
Law of Large Numbers
With the empirical approach, we obtain an approximation but as a procedure is repeated again and again, the empirical probability of an event tends to approach the theoretical probability.
(Imperative) Subjective Probability
a guess or intuition (Ex. "There is a 30% chance of rain showers on Tuesday morning.").
Probability Experiments
an action, or trial, through which specific results are obtained.
Simple Event
an event that has exactly one outcome (Ex. having a baby has one outcome--boy or girl).
Permutations
an ordered arrangement of objects (arranging things in a particular order); ! is a factorial.
Empirical (or Statistical) Probability
based on observations obtained from probability experiments
The Fundamental Counting Principle
can be used to find the number of ways two or more events can occur in sequence; if one event can occur in m ways and a second can occur in n ways, then the number of ways the two events can occur in sequence is mn (this rule can be extended to any number of events).
Always express a probability as a __ or __ number between 0 and 1.
fraction; decimal.
Tree Diagram
gives a visual display of the outcomes of a probability experiment by using branches that originate from a starting point.
Independent Events
if the occurrence of one event does not affect the probability of the other.
Sampling with replacement: selections are __ events, while sampling without replacement: selections are __ events.
independent; dependent.
Complementary Events
the complement of event E, denoted by E' or E (with line over it), is the set of all outcomes in a sample space that are not included in event E.
Combinations
the number of ways to choose r objects from n objects without regard to order.
Conditional Probability
the probability of an event occurring, given that another event has already occurred; denoted by P(B/A).
Outcome
the result of a single trial in a probability experiment.
Sample Space
the set of all the possible outcomes in a probability experiment; written in {}.
Event
the subset of the sample space.
The Multiplication Rule
used for finding P(A and B), the probability that event A occurs in a first trial and event B occurs in a second trial.
Classical Approach
used when each of our outcome in the sample is equally likely to occur.
How could we easily identify the sample space for the last probability experiment?
using a tree diagram or a chart.
Mutually Exclusive (or Disjoint)
when A and B cannot occur at the same time.
The Addition Rule
where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.
Possible Vales for Probabilities run from... For any event E, the probability of E is between __ and __ inclusive.
zero to one; 0; 1
