Week 13 - Rules of probability, and probability distributions Pt1 Flashcards
What is probability?
the measure of the likelihood or chance that a given event will occur. It is a number between 0 and 1,
where:
0 means the event will not happen.
1 means the event will definitely happen
What is the counting rule for multiple-step experiments?
a method to determine the total number of possible outcomes in an experiment that consists of a sequence of steps, where each step has a certain number of possible outcomes. This rule applies to experiments where the outcome of each step is independent of the others.
If an experiment has k steps, and each step has a different number of possible outcomes, the total number of possible outcomes for the entire experiment is the product of the number of outcomes at each individual step.
In mathematical terms, if:
Step 1 has n_1 possible outcomes,
Step 2 has n_2possible outcomes,
Step 3 has n_3 possible outcomes,
and so on, until Step 𝑘
hen the total number of outcomes for the experiment is given by:
Totaloutcomes = n_1 x n_2 x … x n_k
What is an example of the counting rule for multiple-step experiments?
Imagine you’re deciding on a meal. The experiment consists of 3 steps:
Step 1: Choose a type of main dish (3 options: pizza, pasta, or salad).
Step 2: Choose a drink (2 options: water or soda).
Step 3: Choose a dessert (2 options: cake or ice cream).
Using the counting rule, the total number of possible meal combinations is:
Total outcomes = 3 x 2 x 2 = 12
What helps visualise the counting rule for multiple step experiments?
A helpful graphical representation of a multiple-step experiment is a tree diagram.
What is the counting rule for combinations?
number of combinations of N objects taken
n at a time is a counting rule used when you need to choose n objects from a set of
N objects, where the order of selection does not matter.
often referred to as combinations and is denoted as
(N)
(n)
which is read as “N choose n.”
What is the counting rule for combinations equation?
. (N) (N)
𝐶(𝑛) = (n) = N!/ n!(N - n)!
N! = N(N-1)(N-2)…(2)(1)
n! = n(n-1)(n-2)…(2)(1)
0! = 1
Counting rule for combinations example:
A quality control inspector randomly selects 2 out 5 five parts to test for defects.
In a group of 5 parts (A, B, C, D, E), how many combinations of two parts can be selected?
N = 5 and n = 2
. (5) (5)
𝐶(2) = (2) = 5!/ 2!(5 - 2)! = 5x4x3x2x1/ (2x1)(3x2x1)
= 120/12 = 10
10 combinations:
AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE
What is the counting rule for permutations?
used to determine how many ways you can arrange or order a set of objects.
count the number of experimental outcomes when n objects are to be selected from a set of N objects, where the order of selection is important
What is the equation for counting rule for permutations?
. (N) (N)
P(𝑛) = (n) = N!/ (N - n)!
N! = N(N-1)(N-2)…(2)(1)
n! = n(n-1)(n-2)…(2)(1)
0! = 1
Counting rule for permutations:
A quality control inspector randomly selects 2 out 5 five parts to test for defects.
In a group of 5 parts (A, B, C, D, E), how many permutations of two parts can be selected?
Counting rule for combinations:
N = 5 and n = 2
. (5) (5)
P(2) = 2!(2) = 5!/(5 - 2)! = 5x4x3x2x1/ (3x2x1) =
= 120/6 = 20
20 Permutations: AB, BA, AC, CA, AD, DA, AE, EA,BC, CB, BD, DB, BE, EB, CD, DC, CE, EC, DE, and ED
What are 3 approches to assign probabilities?
- classical method
- relative frequency method
- subjective method
What is the classical method?
Assigning probabilities based on the the assumption that all outcomes in a sample space are equally likely
If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome.
Experiment: Rolling a die
Sample Space: S = {1, 2, 3, 4, 5, 6}
Probabilities: Each sample point has a 1/6 chance of occurring
What is the relative frequency method?
assigns probabilities based on historical data or experimentation. In this approach, the probability of an event is estimated by the relative frequency of the event occurring in repeated trials or past occurrences
What is the relative frequency method formula?
P(A) = number of times event A occurs/ total number of trials or observations
What is the subjective method?
assigning probabilities is based on personal judgment, experience, or belief. Rather than relying on mathematical models or historical data, the probability is assigned based on an individual’s opinion about how likely an event is to happen.
Why does the subjective method not use historical data?
When economic conditions and a company’s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data.
We can use any data available as well as our experience and intuition, but ultimately a probability value should express our degree of belief that the experimental outcome will occur.
What is the best method in assigning probabilities
The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate.
What is an event?
a collection of sample points
What is the probability of any event equal to?
equal to the sum of the probabilities of the sample points in the event
How can we compute the probability of an event?
If we can identify all the sample points of an experiment and assign a probability to each
Event and their probabilities example?
1. Event M = Marrley Oil Profitable
M = {(10, 8), (10, -2), (5, 8), (5, -2)}
2. Event C = Cullins Mining Profitable
C = {(10, 8), (5, 8), (0, 8), (-20, 8)}
- P(M) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2)
= 0.20 + 0.08 + 0.16 + 0.26
= 0.70 - P(C) = P(10, 8) + P(5, 8) + P(0, 8) + P(-20, 8)
= 0.20 + 0.16 + 0.10 + 0.02
= 0.48
What are the 4 basic relationships of probability?
- complement of an event
- union of two events
- intersection of two events
- mutually exclusive events
What is the complement of an event?
The complement of event A is defined to be the event consisting of all sample points that are not in A.
The complement of A is denoted by Ā or A’
venn digram in the circle is A, outside the circle not A so A’
What is the union of two events?
The union of events A and B is the event containing all sample points that are in A or B or both.
The union of events A and B is denoted by 𝐴∪𝐵
Union of two events example:
Event M = Marrley oil profitable
Event C = Cullins mining profitable
M ∪ C = Marrley oil profitable or Cullins mining profitable
M ∪ C = {(10, 8), (10, -2), (5, 8), (5, -2), (0, 8), (-20, 8)}
P(M ∪ C) = P(10, 8) + P(10, -2) + P(5, 8) + P(5, -2)
+ P(0, 8) + P(-20, 8)
= 0.20 + 0.08 + 0.16 + 0.26 + 0.10 + 0.02
= 0.82
What is the intersection of two events
The intersection of events A and B is the set of all sample points that are in both A and B.
The intersection of events A and B is denoted by 𝐴∩𝐵.
Intersection of two events example:
Event M = Marrley oil profitable
Event C = Cullins mining profitable
𝑀∩𝐶 = Marrley Oil Profitable
and Cullins Mining Profitable
𝑀∩𝐶 = {(10, 8), (5, 8)}
P(𝑀∩𝐶) = P(10, 8) + P(5, 8)
= 0.20 + 0.16
= 0.36
What is addition law?
provides a way to compute the probability of event A, or B, or both A and B occurring.
P(A∪B) = P(A) + P(B) - P(A∩B)
Addition law example:
Event M = Marrley oil profitable
Event C = Cullins mining profitable
M ∪ C = Marrley oil profitable or Cullins mining profitable
P(M ∪ C) = P(M) + P(C) - P(𝑀∩𝐶)
= 0.7 + 0.48 - 0.36
= 0.82
This result is the same as that obtained earlier using the definition of the probability of an event.
What are mutually exclusive events?
if the events have no sample points in common.
Two events are mutually exclusive if, when one event occurs, the other cannot occur
like in a venn diagram circle A doesnt touch circle B
What is the mutually exclusive formula
P(A∪B) = P(A) + P(B)
What is conditional probability?
The probability of an event given that another event has occurred
The conditional probability of A given B is denoted by P(A|B)
What is the conditional probability equation?
P(A|B) = P(A∩B)/ P(B)
Conditional probability example:
Event M = Marrley oil profitable
Event C = Cullins mining profitable
P(C|M) = Cullins mining profitable given Marrley oil profitable
P(C∩M) = 0.36, P(M) = 0.7
P(C|M) = P(C∩M)/ P(M)
= 0.36/ 0/7 = 0.5143
What is multiplication law?
provides a way to find the probability of the intersection of two events.
It helps to calculate the probability that two or more events will occur together (i.e., the probability of both events happening at the same time).
What is the multiplication law equation?
P(A∩B) = P(B) P(A|B)
Multiplication law example:
Event M = Marrley oil profitable
Event C = Cullins mining profitable
M ∩ C = Marrley Oil Profitable
and Cullins Mining Profitable
We know: P(M) = 0.70, P(C|M) = 0.5143
Thus: P(M ∩ C) = P(M)P(C|M)
= (0.70)(0.5143)
= 0.36
This result is the same as that obtained earlier using the definition of the probability of an event.
What are independent events?
If the probability of event A is not changed by the existence of event B, we would say that events A and B are independent.
What are the formulas for independent events?
P(A|B) = P(A) or P(B|A) = P(B)
How can the multiplication law be used for independent events?
The multiplication law also can be used as a test to see if two events are independent.
P(A ∩ B) = P(A)P(B)
Multiplication law for independent events example:
Event M = Marrley Oil Profitable
Event C = Cullins Mining Profitable
We know: P(M∩C) = 0.36, P(M) = 0.70, P(C) = 0.48
P(M)P(C) = (0.70)(0.48) = 0.34, not 0.36
Hence: M and C are not independent.
Are mutually exclusive events independent of each other?
Two events are mutually exclusive if they cannot occur at the same time. In other words, if event A occurs, event B cannot occur, and vice versa.
Two events are independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, the probability of both events occurring together is the product of their individual probabilities.
For mutually exclusive events, if one event occurs, the probability of the other event occurring is 0 (since they cannot happen together).
However, for independent events, the probability of both events occurring together is the product (times) of their individual probabilities
Therefore, mutually exclusive events cannot be independent because the occurrence of one event directly affects the probability of the other event occurring (it makes the probability of the other event 0).
