Probability and discrete

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STAM4000
Quantitative Methods
Week 3
Probability and discrete
probability distributions
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Understand basic laws of probability
Construct discrete probability
distributions:
Discrete random
variable, quantitative or numerical
(numbers with units) that is FINITE
and COUNTABLE.
Classify probability distributions
Week 3
Probability an
discrete probabi
distributions
Learning
Outcomes

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Why does this
matter?
Probability is the
first step in
quantifying the
possibility of
something
happening or not
happening …
Photo by Benjamin Davies on Unsplash
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#1 Understand basic laws of probability
https://www.pinterest.com.au/pin/524810162798748661/
What is probability?
In general, probability is the chance that something will happen.
Most events are uncertain: probability theory is a way of quantifying the uncertainty.
In any case, a probability is a number which must obey certain rules.
The most basic rules of probability:
• 0 ≤ P(x) ≤ 1
• ∑ P(x) = 1
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#1 Law of Large Numbers versus the so-called Law of Averages
https://www.pinterest.com.au/pin/528961918705000163/
Law of Large Numbers:
If we repeat an
experiment independently,
many times,
the relative frequency of
the event
gets closer to a
single value.
So-called Law of Averages:
everything in
the
short-run
must even out
– this is FALSE.
https://tvtropes.org/pmwiki/pmwiki.php/Main/GamblersFallacy
Law of large numbers
If the events are independent, then the long-run relative frequency of an event gets closer and
closer to a single value, as the number of trials increases. Example: If a gambler plays many times
and loses money, then they will continue to lose money. The gambler’s behaviour follows the Law
of large numbers.
Don’t confuse the Law of large numbers with the so-called Law of averages.
The Law of averages says that things must even out in the short run, that is, that random
phenomena
compensate for whatever happened in the past. This is just not true. The Law of
averages does NOT exist.
Gambler’s fallacy:
When an individual is addicted to gambling, when they lose money, then they will continue to lose
money.
The gambler incorrectly believes in the Law of averages – that they will eventually start to win –
this is false.
This is called the Gambler’s Fallacy. Watch: https://www.youtube.com/watch?v=K8SkCh-n4rw
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Events are INDEPENDENT, if one event does NOT
influence or affect the occurrence of another
event.
Week 1, relative frequency was the proportion,
fraction or percentage RELATIVE to the TOTAL.
Second cartoon
refers to the
“Gambler’s
Fallacy” follows
the fake Law of
averages.
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For all valid probability distributions:
P(x) is read as the “probability of x”.
0 ≤ P(x) ≤ 1
This means the lowest
probability is 0 and the
highest probability is 1.
∑ P(x) = 1
The probability of all possible
outcomes must sum to 1.
Basic rules of probability Distributions or collections or sets
∑ is capital or uppercase sigma.
∑ is read as “SUM OF”
Notation
We usually represent events with a letter e.g.; X
P(x) is read as “the probability of outcome x from all possible events of the random variable X”.
Sometimes P(x) is written as p(x) or prob(x).
When dealing with probabilities, if we get a negative probability or if we get a probability
greater than 1, we know that there is an error in our calculations.
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#1 Experiment, outcomes, trial, events & simple probability
Experiment: a process that
produces several
possible outcomes
(sometimes called a random
experiment)
Outcomes (O
1, O2,…, On)
Individual observations
from an experiment
Trial:
one repetition of the
process, e.g., clinical
trial, rolling dice …
Elementary Event:
An event consisting of a single
outcome, i.e., one that cannot be
decomposed or broken down into
other events,
e.g., roll a “1”
with a six- sided dice
Simple probability:
each outcome has the same
chance of occurring. The
probability of an event A is
denoted P(A)
e.g., each side of a
six-sided
dice has a 1
6
chance.
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Experiment, outcomes, trial, events & simple probability
Example: Flipping two coins, 1 time
Possible outcomes HH, HT, TH, TT
Event (T T) = when both coins land on T
Simple probability: each outcome has the same chance of occurring. The probability of an event
A is denoted P(A).
This is theoretical probability.
P(A) =
number of outcomes in A
Total number of possible outcomes
With simple probability, each outcome is equally likely.
Example: Flipping a two-sided coin. Trial: land on tail. Possible outcomes: T, H
Probability of landing on T = ½ = 0.50; Probability of landing on H = ½ = 0.5
Example: Roll a six-sided die (one dice). Trial: each roll. Possible outcomes: 1, 2, 3, 4, 5, 6
Probability of Rolling a “1”
= Prob(1)
= P(1)
= 1/6
Note that each number, 1, 2, 3, 4, 5, 6 has the SAME chance of being rolled, so 1/6 is a simple
probability.
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#1 Basic concepts in probability
Collectively exhaustive events:
all possible events of a trial are
considered.
Contingency
table:
a two-way table
used to
summarise
variables.
Marginal probability:
the probability of a single event,
E.g., if using a contingency table,
P(A) = Row total event A
Table overall total
Complementary
event: opposite to
an event
occurring.