Reading Quiz 6 Flashcards
simulation
the imitation of chance behavior, based on a model that accurately reflects the phenomenon under consideration
random
a phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions
probability of an outcome of a random phenomenon
the proportion of times the outcome would occur in a very long series of repetitions
long-term relative frequency
sample space S
the set of all possible outcomes of a random experiment
element
each individual outcome in the sample space
event
a subset of the sample space
any outcome or a set of outcomes of a random phenomenon
probability model
a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of assigning probabilities to events
three ways to properly enumerate the outcomes in a sample space
- drawing a tree diagram
- using the multiplication counting principle
- making an organized list of all possible outcomes
multiplication counting principle
if you can do one task in m number of ways and a second task in n number of ways then both tasks can be done in m x n number of ways
sampling with replacement
when you pick a card from a deck, for example, and you put it back in the deck before picking your second card
sampling without replacement
if you don’t put the first card back before picking the second
changes the probability for each new selection
if know whether with or without replacement, helps properly identify sample space
a union b
the set of elements which belong to a or b or both (at least one)
upwards u
a intersect b
the set of elements which belong to set a and b (all events occur)
upside down u
empty set
Ø
the event has no outcomes in it
probability
p(a)
always a number between 0 and 1 inclusive
p(a) = 0 p(a) = 1
means event never occurs
means event occurs on every trial
p(s)
s is sample space
p(s) = 1
means that sum of probabilities of all possible outcomes must equal 1
probability of event not occurring
1 - probability that it does occur
p(a^c) = 1 - p(a)
p(a) = 1 - p(a^c)
a^c
a complement
the set of elements that are not in set a but are in the sample space
another notation is a’ or a prime
at least
tip off to think about complement
addition rule for two events
U, union, or
you must first determine if the events are joint or disjoint
if the events are joint: p(a U b) = p(a) + p(b) - p(a intersect b)
if the events are disjoint: p(a U b) = p(a) + p(b)
mutually exclusive (disjoint)
two events are mutually exclusive/disjoint
means that they have no outcomes in common and so can never occur simultaneously
probability that one or the other occurs is the sum of their individual probabilities
joint
opposite of disjoint
means that events can occur simultaneously
multiplication rule for two events
first determine if events are dependent or independent
if events are dependent: p(a intersect b) = p(a) x p(b|a)
if the events are independent: p(a intersect b) = p(a) x p(b)
independent
two events are independent if knowing that one occurs does not change the probability that the other occurs
test for independence
events a and b are independent events if p(b|a) = p(b) if p(a|b) is not equal to p(b) then the events are dependent
don’t confuse
independent events with mutually exclusive (disjoint) events
disjoint events cannot be independent
conditional probability
let a and b be two events with p(a) > 0 then p(b|a) = ( p(a U b) ) / p(a)
symbol p(b|a)
the probability of event B given that event A has occurred
Making a model that accurately reflects the experiment under consideration and imitating chance behavior based on that model is called doing a _____.
simulation
What are the 5 steps of doing simulations?
A. State the problem, state the assumption(s), assign digits to represent outcomes, simulate repetitions, and state the conclusions.
- Someone wants to simulate a situation where there’s a 3/10 chance that a child will be involved in bullying. The person assigns the digits 0 to 3 for involved in bullying, and the rest of the digits to noninvolved in bullying. Do you have a problem with this? If so, what’s your problem?
A. Yes, there is a problem with this. There are 4 digits from 0 to 3 inclusive, and 6 other digits, so the person would be simulating a 40% probability situation rather than a 30% probability situation.
The branch of mathematics that deals with the pattern of chance outcomes is ____.
probability
- The big idea of the study of probability is that chance behavior is unpredictable in the _____ but has a regular and predictable pattern in the _____.
short run, long run
- An illustration of the “big idea” mentioned in #5 is that while it is unpredictable whether a single coin toss will come out heads, the ________ is almost always very close to .5.
A. Percent of heads in a very large number of tosses
- The ____ of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions, i.e. long-term relative frequency.
probability
When there are independent trials, that means that the outcome of one trial _______.
A. Does not influence the probability of another.
- The set of all possible outcomes of a random phenomenon is called the ______.
sample space
An event is defined as a subset of ____.
the sample space
When we make a mathematical description of a random phenomenon by describing a sample space and a way of assigning probabilities to events, we are constructing a
probability model
- Jane has 2 shirts and 3 pairs of pants. If we want to picture the 6 ways she can dress in these garments, we can draw a diagram with a bifurcation point at the left of the page, with two lines going out to two points called “red shirt” and “brown shirt.” From each of these, you then draw 3 lines, saying “blue pants,” “green pants,” and “black pants.” This sort of picture is called a _____.
tree diagram
- Jane has 2 shirts and 3 pairs of pants. The “Cartesian Product” of these two sets produces 6 possible combinations. This illustrates what our book calls the _____ principle, which says that if you can do one task in “a” ways, and another in “b” ways, you can do both together in _____ ways.
ab
- Referring to the first property of probabilities: the probability of any event A has to satisfy the inequality
x P(A) y. What are the values of x and y?
0 and 1
If you sum the probabilities for each member of the sample space, you always come to a grand total of ____.
1
- If the probability that A will occur is P(A), the probability that A will not occur is ____.
A. 1-P(A).
- What notation do we use to represent the probability that A will not occur, if P(A) is the probability that A will occur?
A. P(Ac) or P(A’)
What does it mean when we call two events disjoint or mutually exclusive?
A. That if one happens, the other can’t also happen.
- If two events are mutually exclusive, and P(A) is the probability of A, and P(B) is the probability of B, what
is the probability of (A and B) (a.k.a. A intersect B, a.k.a A B)?
0
- If two events are mutually exclusive, and P(A) is the probability of A, and P(B) is the probability of B, what is the probability of (A or B) (a.k.a. A union B, a.k.a A B)?
A. P(A) + P(B).
Suppose a random event has k equally likely outcomes. What’s the probability of any one of these outcomes?
1/k
- In a finite sample space, with outcomes that are not equally likely, the probability of any event is the sum of the probabilities of the outcomes making up the event. Someone’s statistics instructor asks, “Please give an example of this.” The person says, “Suppose the probability that a randomly selected person in a certain community will own a dog is .3, a cat is .2, and another pet is .1. Then the probability that the person will own a pet is (.3 +.2 +.1).” Do you have a problem with this example? If so, what’s your problem?
A. Yes, one big problem is: what about the people who own two or more different types of pets? If the only people who owned cats or other pets were the dog owners, the probability of pet ownership would be .3 instead of .6. The addition rule only applies to disjoint events, and these categories are not disjoint.
When two events are independent, what is the probability that both will occur?
A. The product of their individual probabilities.
What is the general addition rule for unions of two events that may or may not be disjoint?
A. P(A or B) = P(A) +P(B) –P(A and B).
What does the symbol P(B|A) mean?
A. The probability of B given A, or the probability that B will happen given that A has happened.
What is the general multiplication rule for any two events?
A. P(A and B) = P(A)P(B|A)
Why does this rule simplify to the multiplication rule for independent events, when the two events are independent?
A. Because the definition of independence is that P(B|A) =P(B), another way of saying that A doesn’t influence the occurrence of B. Thus when the two events are independent, we can substitute P(B) for P(B|A) in the general multiplication rule, and get P(A and B)= P(A)P(B).
The general multiplication rule for any two events really follows from the definition of conditional probability, and is a rearrangement of the defining formula. What is the defined formula for conditional probability?
A. P(B|A) = P(A and B) / P(A)
True or false: P(A and B and C) = P(A)P(B|A)P(C|A and B)?
true
True or false: P(A and B and C)=P(A)P(B|A)P(C|A and B) is the basis of tree diagrams.
true
For a tree diagram: the probability of reaching the end of any complete branch is the product of what?
A. The probabilities written on its segments (provided that those probabilities are written so as to mean the conditional probability of going down this branch given that you have reached the previous!)
What should someone do to organize the information for decision analysis if one is given a set of complex information about branching alternatives and outcomes given those alternatives?
use a tree diagram
