What is Probability?
“Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity.”
https://openstax.org/books/introductory-statistics/pages/3-1-terminology
Why is understanding probability in statistics important?
We use probability in our everyday lives to make choices. Should I cross the street on a red light? Or, should I spend time working on my homework, or studying?
https://i.pinimg.com/originals/f0/d7/21/f0d721b00d5433f3ba22bb7dad20af54.jpg
In statistics probability is a formalized system that helps us understand the certainty of outcomes of experiments or activities.
The p-value, (Please LINK TO DEFINITION) a measure of probability, is also used in testing hypotheses (link “hypotheses” to section on hypothesis testing). We take information from sample statistics and generalize about larger populations.
The sample space is the collection of all possible outcomes of an experiment. The outcomes are commonly expressed using one of the four modes illustrated below.
https://openstax.org/books/statistics/pages/3-1-terminology
The p-value is the probability that you would obtain the answer you have, assuming the occurrence is random.
How do we measure probability?
We measure probability using the p-value. Where:
0 = no chance of an event happening, and
1 = the event is certain to happen.
https://openstax.org/books/statistics/pages/3-1-terminology
How do we represent probabilities?
We can represent probabilities as a fraction, a decimal or a percentage. Certainty ranges from 0 (no likelihood of an event occurring) to 1, or 100% (complete certainty).
For example, what is the probability of each number on a six-sided die turning up when rolled?
|
Number on die |
1 |
2 |
3 |
4 |
5 |
6 |
|
Probability expressed as a fraction |
1 / 6 |
1 / 6 |
1 / 6 |
1 / 6 |
1 / 6 |
1 / 6 |
|
Probability expressed as a decimal |
0.167 |
0.167 |
0.167 |
0.167 |
0.167 |
0.167 |
|
Probability expressed as a percent |
16.7% |
16.7% |
16.7% |
16.7% |
16.7% |
16.7% |
Methods for Calculating Probability
Empirical / Relative Frequency is calculated from experiment results.
It is expressed as:
P (A) = number of times A occurred
total number of observations
For example: Flip a coin 50 times.
You get: 23 heads & 27 tails.
The empirical probability of heads turning up is:
23 / 50.
https://www.pngkey.com/detail/u2q8a9e6e6i1w7q8_coin-toss-heads-or-tails-coin-flip-png/
Theoretical probability is calculated when you use your powers of deduction to determine probability.
When all outcomes are equally likely we can easily calculate the theoretical probability of an event.
It is expressed as: P (A) = number of outcomes of event A
total number of outcomes
For example: Flip a coin 50 times.
There are two equally likely results, heads or tails (50%, ½, 0.5)
The theoretical probability of getting heads is half of 50, or 25.
This law states that the more often your repeat an experiment the closer the empirical / relative frequency comes to the theoretical probability.
Can we test this theory? Flip a coin. Can we embed the link with the flippable coin?? If not we can jut keep linked word)
Do it 10 times.
Now do it 25 times.
Now do it 100 times.
Now do it 1000 times… NO? Too much?
https://www.britannica.com/science/law-of-large-numbers
How many times did heads turn up? What about tails? Do your results support the theory?
It can take many tries for the fraction of heads to approach 50%, as illustrated in the image on the commemorative stamp, above.
Independent Events exist when the probability of one outcome does not affect another.
Example: We are rolling 2 dice. We want to know the probability of rolling a 6 on each.
The probability of rolling a 6 on each die is 1/6. To calculate the probability of rolling a six on both we multiply.
P (a and b) = P(a) x P(b)
= 1/6 x 1/6
= 1/36 (or 0.028, or 2.8%)
https://www.youtube.com/watch?v=Azy6d3IEeek
Dependent Events exist when probability events are related
Example: We pull out a blue then a red marble from a bag of 20 marbles where there are 5 green, 5 red, 5 yellow and 5 blue.
P (a) = 5 / 20
P (a then b) = p (a) * p (b given a)
= ¼ * 5/ 19
= 5 / 76
= 0.066 or 6.6% https://www.shutterstock.com/search/marble+bag
Watch out!
Sometimes mistaken beliefs lead us to misinterpret the world around us.
The Gambler’s Fallacy is the mistaken belief that a streak of bad luck makes a person “due” for a streak of good luck, or that a streak of good luck will continue. (Bennett, Briggs and Triola, 2018, p. 209)