Probability Topics by Kathleen DiGirolamo

Probability

kdigirolamo — 2020-06-20 12:03:18 UTC

"a measure of how certain we are of experiment or activity outcomes" (Lemieux, 2018)
- denoted P(event name)
- long-term relative frequency of outcome
- always between zero & one, inclusive 0 <=P(A)<=1
-P(A) = 0 event A can never happen
- P(A) = 1 event A always happens
- Calculation when events are equally likely- count the number of outcomes for event A and divide by the number of outcomes in the sample space
P(A) = outcomes of A / outcomes of Sample Space

Experiment

kdigirolamo — 2020-06-20 12:04:10 UTC

"a planned operation under controlled conditions" (Lemieux, 2018)

Chance

kdigirolamo — 2020-06-20 12:04:56 UTC

"result of an experiment is not predetermined" (Lemieux, 2018)

Outcome

kdigirolamo — 2020-06-20 12:05:18 UTC

"result of an experiment" (Lemieux, 2018)

Sample Space

kdigirolamo — 2020-06-20 12:05:32 UTC

"set of all possible outcomes" (Lemieux, 2018)
- list of outcomes
- tree diagram
- Venn diagram
- S denotes sample space

Event

kdigirolamo — 2020-06-20 12:06:50 UTC

"any combination of outcomes considered a subset of the sample space" (Lemieux, 2018) and represented by uppercase letters

Likelihood of Events

kdigirolamo — 2020-06-20 12:07:42 UTC

Likely - "an event that has a good chance of happening with probability greater than 10%" (Lemieux, 2018)

Unlikely - "a rare event with probability of less than 1%" (Lemieux, 2018)

Equally Likely - "each outcome of an experiment occurs with equal probability" (Lemieux, 2018)
Fair/Non Biased - "outcomes of event are equally likely to occur" (Lemieux, 2018)

Not Equally Likely - "outcomes are likely not to occur with equal probability" (Lemieux, 2018)
Unfair/Biased - outcomes of events may be affected by some factor

Law of Large Numbers

kdigirolamo — 2020-06-20 12:50:58 UTC

"with increased repetitions of an experiment, relative frequency becomes closer to theoretical probability" (Lemieux, 2018)

"OR" Event

kdigirolamo — 2020-06-20 12:53:46 UTC

"an outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B" (Lemieux, 2018)

Example:
Let A = {1,2,3,4,5}, Let B = {4,5,6,7,8}
A OR B = {1,2,3,4,5,6,7,8)
DO NOT Repeat outcomes

"AND" Event

kdigirolamo — 2020-06-20 13:06:13 UTC

"an outcome is in event A AND B if the outcome is in both A and B at the same time" (Lemieux, 2018)

Example:
Let A = {1,2,3,4,5}, Let B = {4,5,6,7,8}
A AND B = {4,5}

Complement

kdigirolamo — 2020-06-20 13:10:16 UTC

-all outcomes not in an event
-denoted A', ("A prime") for event A
-P(A) + P(A') = 1

Example:
Let S = {1,2,3,4,5,6}
Let A = {1,2,3,4}, Then A' = {5,6}
P(A) = 4/6, P(A') = 2/6
P(A) + P(A') = 4/6 + 2/6 = 1

Conditional Probability

kdigirolamo — 2020-06-20 13:16:08 UTC

-"probability that event A will occur given that event B has already occurred" (Lemieux, 2018), written P(A|B)
-"conditional reduces the sample space" (Lemieux, 2018)

Calculation Formula:
P(A|B) = P(A AND B) / P(B)

Odds

kdigirolamo — 2020-06-20 13:25:26 UTC

"probability of events as a ratio of success to failure" (Lemieux, 2018)
P(A) - probability of success
1 - P(A) - probability of failure
read numerator to denominator
Mathematically:
P(A) / 1 - P(A)

Problem Solving

kdigirolamo — 2020-06-20 13:28:55 UTC

-read wording CAREFULLY
-understand events
-clearly identify the event of interest
-determine presence of a condition
-identify that condition

Reference

kdigirolamo — 2020-06-20 13:37:23 UTC

Lemieux, C. (2018, July 31). Business Statistics - Unit 2 - Probability. OpenStax CNX. Retrieved from https://cnx.org/contents/26zux3nL@3.7:hl6XXT3P@3/Terminology-Probability-Topics-MtRoyal-Version2016RevA

Independent Events

kdigirolamo — 2020-06-20 13:39:32 UTC

"A and B are independent if one occurrence does not affect the chance the other occurs" (Lemieux, 2018)

To be independent, one must show at least one of the following:
P(A|B) = P(A)
P(B|A) = P(B)
P(A AND B) = P(A)P(B)

Dependent events are not independent and the above conditions do not exist.

"Assume A and B are dependent until proven otherwise." (Lemieux, 2018)

Example: Let event A = learning Spanish. Let event B = learning German. Then A AND B = learning Spanish and German. Suppose P(A) = 0.4 and P(B) = 0.2. P(A AND B) = 0.08. Are events A and B independent? (Lemieux, 2018)

P(A AND B) = P(A)P(B)
0.08 = 0.4 x 0.2
P(A AND B) are equal to P(A)P(B), therefore the events are independent.

Mutually Exclusive Events

kdigirolamo — 2020-06-20 13:39:55 UTC

"events that cannot occur at the same time" (Lemieux, 2018)

A and B do not share any outcomes
P(A OR B) = P(A) + P(B) and
P(A AND B) = 0

Example:
S={1,2,3,4,5,6,7,8,9,10}
Let A={1,2,3,4,5}, B={4,5,6,7,8}, C={7,9}
A AND B = {4,5}, P(A AND B)=2/10
which does not equal zero. A AND B are NOT mutually exclusive. (Lemieux, 2018)

A AND C have no numbers in common so P(A AND C) = 0, so
A and C ARE mutually exclusive.

"Assume A and B are mutually exclusive until proven otherwise."
(Lemieux, 2018)

Sampling

kdigirolamo — 2020-06-20 13:44:14 UTC

With Replacement - "members of a population are replaced after it's picked, then it can have the possibility of being chosen more than once." (Lemieux, 2018)
The result of the first pick will NOT affect the second which is deemed INDEPENDENT

Without Replacement - "each member of a population may be chosen ONLY once." (Lemieux, 2018)
Probability of the second pick will be affected by the result of the first pick with results deemed DEPENDENT

Reference

kdigirolamo — 2020-06-20 14:05:44 UTC

Lemieux, C. (2018, July 31). Business Statistics - Unit 2 - Probability. OpenStax CNX. Retrieved from https://cnx.org/contents/26zux3nL@3.7:ImPTSfLP@2/Independent-and-Mutually-Exclusive-Events-Probaility-Topics-MtRoyal-Version2016RevA

Calculating Probabilities

kdigirolamo — 2020-06-20 14:10:23 UTC

The Multiplication Rule and the Addition Rule are 2 rules to think about to "determine if 2 events are independent or dependent and if they are mutually exclusive or not." (Lemieux, 2018)

Multiplication Rule ("and")

kdigirolamo — 2020-06-20 14:14:18 UTC

P(A AND B) = P(B)P(A|B) rewritten:
P(A|B) = P(A AND B) / P(B), "read the probability of A given B is equal to the probability of A and B divided by the probability of B." (Lemieux, 2018)

If A and B are independent, then
P(A|B) = P(A)

" "and" means that the event has to satisfy TWO conditions." (Lemieux, 2018)

So, "multiplying fractions would yield a smaller result indicating the increasing dificulty of satisfying 2 conditions." (Lemieux, 2018)

Addition Rule ("or")

kdigirolamo — 2020-06-20 14:23:23 UTC

P(A OR B)=P(A) + P(B) - P(A AND B)

If A and B are mutually exclusive, then P(A AND B) = 0, so
P(A OR B) = P(A) + P(B)
(Lemieux, 2018)

Examples of Two Basic Rules

kdigirolamo — 2020-06-20 14:30:35 UTC

"Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event that Carlos is successful on his 1st attempt. P(A) = 0.65. B = the event Carlos is successful on his second attempt. P(B) = 0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal | that he made the first goal is 0.90.

What is the probability he makes both goals?" (Lemieux, 2018)

(multiplication rule)
P(A AND B) = P(A)P(B|A)
P(A AND B) =( 0.65)(0.9)
P(A AND B) = 0.585

What is the probability that Carlos makes either the first goal or the second goal? (Lemieux, 2018)

(addition rule)
P(A OR B)=P(A) + P(B) - P(A AND B)
P(A OR B)=(0.65 + 0.65) - 0.585
P(A OR B) = 0.715

Reference

kdigirolamo — 2020-06-20 14:38:40 UTC

Lemieux, C. (2018, July 31). Business Statistics - Unit 2 - Probability. OpenStax CNX. Retrieved from https://cnx.org/contents/26zux3nL@3.7:JFwiPFqf@12/Two-Basic-Rules-of-Probability

Contingency Tables

kdigirolamo — 2020-06-20 14:45:41 UTC

"A display of sample values related to 2 variables that may be dependent or contingent upon one another." (Lemieux, 2018)

A contingency table eases the determination of conditional probabilities.

Rows and columns total with Total row = Total column = sample space

Example Contingency Table with Questions: (Lemieux, 2018)

What is P(athlete stretches)?
P(athlete stretches) = 350/800

What is P(athlete stretches|no injury)?
P(athlete stretches|no injury) =
295/514

Tree Diagrams

kdigirolamo — 2020-06-20 14:59:15 UTC

Used with complex probability problems as a visualization and aide to solutions.

"Type of graph used to determine the outcomes of an experiment." (Lemieux, 2018)

Has "branches labeled with either frequencies of outcome or probabilities." (Lemieux, 2018)

"The first draw is shown in the first set of branches, the second draw in the second set of branches." (Lemieux, 2018)

Outcomes of branches are products of first and second draw.

Different outcomes generated "with replacement" and "without replacement" events.
(Lemieux, 2018)

Tree Diagram Example with Replacement

kdigirolamo — 2020-06-20 15:14:56 UTC

"In a 52 card deck, there are 12 face cards (event F) and 40 not face cards (event N). Draw two cards, one at a time, with replacement. All possible outcomes are shown in the tree diagram as frequencies. Calculate P(FF)." (Lemieux, 2018)
P(FF) = 144/2704 = 0.053

Tree Diagram Example without Replacement

kdigirolamo — 2020-06-20 15:26:35 UTC

"A litter of kittens includes four tabby kittens and five black kittens. Two kittens are randomly selected for adoption without replacement." (Lemieux, 2018)

What is the probability that both kittens are tabby?
P(TT) = (4/9)(3/8) = 12/72 = 1/6

What is the probability that one kitten of each color is selected?
P(TB) = P(B|T first) + P(T|B first)
P(TB) = (4/9)(5/8) + (5/9)(4/8)
= 20/72 + 20/72 = 40/72

Reference

kdigirolamo — 2020-06-20 15:41:04 UTC

Lemieux, C. (2018, July 31). Business Statistics - Unit 2 - Probability. OpenStax CNX. Retrieved from https://cnx.org/contents/26zux3nL@3.7:pOnG81VU@2/Contingency-Tables-and-Tree-Diagrams-Probability-Topics-MtRoyal-Version2016RevA