Language isn’t very precise in dealing with uncertainty. Probably is more certain than possibly, but who knows where dollars-to-doughnuts falls on the spectrum. Statistics needs to deal with uncertainty more quantitatively. That’s where probability comes in. For the most part, probability is a simple concept with fairly straightforward formulas. The applications, however, can be mind bending.
Where Do Probabilities Come From?
The first question you may have is “were do probabilities come from?” They sure don’t come from arm-waving know-it-alls at work or anonymous memes on the internet. Real probabilities come from four sources:
Logic – Some probabilities are calculated from the number of logical possibilities for a situation. For example, there are two sides to a coin so there are two possibilities. Standard dice have six sides so there are six possibilities. A standard deck of playing cards has 52 cards so there are 52 possibilities. The formulas for calculating probabilities aren’t that difficult either. By the time you finish reading this blog, you’ll be able to calculate all kinds of probabilities.
Data – Some probabilities are based on surveys of large populations or experiments repeated a large number of times. For example, the probability of a random American having a blood type of A-positive is 0.34 because approximately 34% of the people in the U.S. have that blood type. Likewise, there are probabilities that a person is a male (0.492), a non-Hispanic white (0.634), having brown eyes (0.34) and brown hair (0.58). The internet has more data than you can imagine for calculating probabilities.
Oracles – Some probabilities come from the heads of experts. Not the arm-wavers on the internet, but real professionals educated in a data-driven specialty. Experts abound. Sports gurus live in Las Vegas, survey builders reside largely in academia, and political prognosticators dwell everywhere. Some experts make predictions based on their knowledge and some use their knowledge to build predictive models. Probabilities developed from expert opinions may not all be as reliable as probabilities based on logic or data, but we rely heavily on them.
Models – A great many probabilities are derived from mathematical models, built by experts from data and scientific principles. You hear meteorological probabilities developed from models reported every night on the news, for instance. These models help you plan your daily life, so their impact is great. Perhaps even more importantly, though, are the probabilistic models that serve as the foundation of statistics itself, like the Normal distribution.
Turning Probably Into Probability
Probabilities are discussed in terms of events or alternatives or outcomes. They all refer to the same thing, something that can happen. A few basic things you need to know about the probability of events are:
- The probability of any outcome or event can range only from 0 (no chance) to 1 (certainty).
- The probability of a single event is equal to 1 divided by the number of possible events.
- If more than one event is being considered as favorable, then the probability of the favorable events occurring is equal to the number of favorable events divided by the number of possible events.
These rules are referred to as simple probability because they apply to the probability of a single independent, disjoint event from a single trial. (Independent and disjoint events are described in the following paragraphs.) A trial is the activity you preform, also called a test or experiment, to determine the probability of an event. If the probability involves more than one trial, it is called a joint probability. Joint probabilities are calculated by multiplying together the relevant simple probabilities. So, for example, if the probability of event A is 0.3 (30%) and the probability of event B is 0.7 (70%), the joint probability of the two events occurring is 0.3 times 0.7, or 0.21 (21%). The probability of a brown-eyed (0.34), brown-haired (0.58), non-Hispanic-white (0.63) male (0.49) having A-positive blood (0.34), for instance, is only 2%, quite rare. Joint probabilities also range only from 0 to 1.
Events can be independent of each other or dependent on other events. For example, if you roll a dice or flip a coin, there is no connection between what happens in each roll or flip. They are independent events. On the other hand, if you draw a card from a standard deck of playing cards, your next draw will be different from the first because there are now fewer cards in the deck. Those two draws are called dependent events. Calculating the probability of dependent events has to account for changes in the number of total possible outcomes. Other than that, the formula for probability calculations is the same.
Some outcomes don’t overlap. They are one-or-the-other. They both can’t occur at the same time. These outcomes are said to be mutually exclusive or disjoint. Examples of disjoint outcomes might involve coin flips, dice rolls, card draws, or any event that can be described as either-or. For a collection of disjoint events, the sum of the probabilities is equal to 1. This is called the Rule of Complementary Events or the Rule of Special Addition.
Some outcomes do overlap. They can both occur at the same time. These outcomes are called non-disjoint. Examples of non-disjoint outcomes include a student getting a grade of B in two different courses, a used car having heated seats and a manual transmission, and a playing card being a queen and in a red suit. For a collection of non-disjoint events, the sum of the simple probabilities minus the probability in common for the events is equal to 1. This is called the Rule of General Addition. The joint probability of non-disjoint events is called a Conditional Probability.
There is a LOT more to probability than that, but that’s enough to get you through Stats 101. Read through the examples to see how probability calculations work.
You’ve Probably Thought of These Examples
Probability does not indicate what will happen, it only suggests how likely, on a fixed numerical scale, something is to happen. If it were definitive, it would be called certainility not probability. Here are some examples.
Coins. Find a coin with two different sides, call one side A and the other side B.
What is the probability that B will land facing upward if you flip the coin and let it land on the ground?
- Probability = number of favorable events / total number of events
- Probability = 1 / 2
- Probability = 50% or 0.5 or ½ or 1 out of 2.
This is a probability calculation for two independent, disjoint outcomes. Coin edges aren’t included in the total-number-of-events because the probability of flipping a coin so it lands on an edge is much much smaller than the probability of flipping a coin so it lands on a side. Alternative events have to have an observable (non-zero) probability of occurring for a calculation to be valid.
Probability can be expressed in several ways — as a percentage, as a decimal, as a fraction, or as the relative frequency of occurrence.
Using the same coin, record the results of 100 coin-flips. Count the number of times the results were the A-side and how many times the results were the B-side. Then flip the coin one more time.
What is the probability that side B will land facing upward?
- Probability = number of favorable events / total number of events
- Probability = 1 / 2
- Probability = 50% or 0.5 or ½ or 1 out of 2.
Each coin flip is independent of the results of every other coin flip. So, whether you flip the coin 100 times or a million times, the probability of the next flip will always be ½. When you flipped the coin 100 times, for instance, you might have recorded 53 B-sides and 47 A-sides. The probability of the B-side facing upward after a flip would NOT be 53/100 because the flips are independent of each other. What happens on one flip has no bearing on any other flip.
Toast. Make two pieces of toast and spread butter on one side of each. Eat one and toss the other into the air.
What is the probability that the buttered side will land facing upward?
- Probability = number of favorable events / total number of events
- Probability = 1 / 2
- Probability = 50% or 0.5 or ½ or 1 out of 2.
WRONG. You knew this was wrong because the buttered side of a piece of toast usually lands facing down. That’s the result of the buttered side being heavier than the unbuttered side. The two sides aren’t the same in terms of characteristics that will dictate how they will land. For probability calculations to be valid, each event has to have a known, constant chance of occurring. Unlike a coin, the toast is “loaded” so that the heavier side faces downward more often. Now, say you knew the buttered side landed downward 85% of the time. You could calculate the probability that the buttered side of your next toast flip will land facing upward as:
- Probability = number of favorable events / total number of events
- Probability = (100-85) / 100 = 15 / 100
- Probability = 15% or 0.15 or 3/20 or 1 out of 6⅔.
To make this calculation valid, all you would have to do is establish that, when tossed into the air, buttered toast will land with its unbuttered-side upward a constant percentage of the time. So, make 100 pieces of toast and butter one side of each …. Let me know how this turns out.
Standard Dice. Standard dice are six-sided with a number (from 1 to 6) or a set of 1 to 6 small dots (called pips) on each side. The numbers (or number of pips) on opposite sides sum to 7.
What is the probability that a 6 (or 6 pips) will land facing upward when you toss the dice?
- Probability = number of favorable events / total number of events
- Probability = 1 / 6
- Probability = 17% or 0.167 or ⅙ or 1 out of 6.
This is a probability calculation for 6 independent, disjoint outcomes.
What is the probability that an even number (2, 4, or 6) will land face upward when you toss the dice?
- Probability = number of favorable events / total number of events
- Probability = 3 / 6
- Probability = 50% or 0.5 or ½ or 1 out of 2.
This calculation considers 3 sides of the dice to be favorable outcomes.
What is the probability that a 1 will land face upward on two consecutive tosses of the dice?
- Probability = (Probability of Event A) times (Probability of Event B)
- Probability = (1 / 6) times (1 / 6)
- Probability = 3% or 0.28 or ^{1}/_{36} or 1 out of 36.
This calculation estimates the joint probability of 2 independent, disjoint outcomes occurring based on 2 rolls of 1 dice.
What is the probability that, using 2 dice, you will roll “snake eyes” (only 1 pip on each dice)?
- Probability = (Probability of Event A) times (Probability of Event B)
- Probability = (1 / 6) times (1 / 6)
- Probability = 3% or 0.28 or ^{1}/_{36} or 1 out of 36.
This calculation also estimates the joint probability of 2 independent, disjoint outcomes occurring based on 1 roll of 2 dice.
What is the probability that a 6 will land face upward on 3 consecutive rolls of the dice?
- Probability = (Probability of Event A) times (Probability of Event B) times (Probability of Event C)
- Probability = (1 / 6) times (1 / 6) times (1 / 6)
- Probability = 0.5% or 0.0046 or ^{23}/_{5,000} or 1 out of 216.
Roll 1 dice 3 times or 3 dice 1 time, if you get 666 either the dice is loaded or the Devil is messing with you.
DnD Dice. Dice used to play Dungeons and Dragons (DnD) have different numbers of sides, usually 4, 6, 8, 10, 12, and 20.
What is the probability that 6 will land facing upward when you throw a 20-sided (icosahedron) dice?
- Probability = number of favorable events / total number of events
- Probability = 1 / 20
- Probability = 5% or 0.05 or ^{1}/_{20} or 1 out of 20.
This is a probability calculation for 20 independent, disjoint outcomes.
What is the probability that a 6 will land face upward on 3 consecutive tosses of the 20-sided (icosahedron) dice?
- Probability = (Probability of Event A) times (Probability of Event B) times (Probability of Event C)
- Probability = (1 / 20) times (1 / 20) times (1 / 20)
- Probability = 0.01% or 0.000125 or ^{1}/_{8,000} or 1 out of 8,000.
So, you have a smaller chance of summoning the Devil by rolling 666 if you use a 20-sided DnD dice instead of a standard 6-sided dice.
What is the probability that 6 will land facing upward when you throw a 4-sided (tetrahedron, Caltrop) dice?
- Probability = number of favorable events / total number of events
- Probability = 0 / 4
- Probability = 0% or 0.0 or ^{0}/_{4} or 0 out of 4.
There is no 6 on the 4-sided dice. Not everything in life is possible.
Playing Cards. A standard deck of playing cards consists of 52 cards in 13 ranks (an Ace, the numbers from 2 to 10, plus a jack, queen, and king) in each of 4 suits – clubs (♣), diamonds (♦), hearts (♥) and spades (♠).
What is the probability that you will draw a 6 of clubs from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 1 / 52
- Probability = 2% or 0.019 or ^{1}/_{52} or 1 out of 52.
This is a probability calculation for 52 disjoint outcomes. The draw is independent because only one card is being drawn.
What is the probability that you will draw a 6 from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 4 / 52
- Probability = 8% or 0.077 or ^{1}/_{13} or 1 out of 13.
This is a probability calculation for 4 favorable disjoint outcomes out of 52 because there is a 6 in each of the 4 suits.
What is the probability that you will draw a club (♣), from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 13 / 52
- Probability = 25% or 0.25 or ¼ or 1 out of 4.
This is a probability calculation for 13 favorable disjoint outcomes out of 52 because there are 13 club cards in the deck.
What is the probability that you will draw a club (♣), from a partial deck?
You can’t calculate that probability without knowing what cards are in the partial deck.
Tarot Cards. A deck of Tarot cards consists of 78 cards, 22 in the Major Arcana and 56 in the Minor Arcana. The cards of the Minor Arcana are like the cards of a standard deck except that the Jack is also called a Knight, there are 4 additional cards called Pages, 1 in each suit, and the suits are Wands (Clubs), Pentacles (Diamonds), Cups (Hearts), and Swords (Spades).
What is the probability that you will draw a Major Arcana card from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 22 / 78
- Probability = 28% or 0.282 or ^{11}/_{39} or 1 out of 3.54.
This is a probability calculation for 22 disjoint outcomes. The draw is independent because only 1 card is being drawn.
What is the probability that you will draw a Knight from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 4 / 78
- Probability = 5% or 0.051 or ^{2}/_{39} or 1 out of 19.5.
This is a probability calculation for 4 disjoint outcomes. The draw is independent because only 1 card is being drawn.
What is the probability that you will draw Death (a Major Arcana card) from a complete deck?
- Probability = number of favorable events / total number of events
- Probability = 1 / 78
- Probability = 1% or 0.0128 or ^{1}/_{78} or 1 out of 78.
If the Death card turns up a lot more than 1% of the time, maybe seek professional help.
Assuming you have already drawn Death, what is the probability that you will draw either The Tower, Judgement, or The Devil (other Major Arcana cards) from the same deck?
- Probability = number of favorable events / total number of events
- Probability = 3 / 77
- Probability = 4% or 0.039 or ^{3}/_{77} or 1 out of 25.7.
This is a probability calculation for 77 disjoint outcomes. The draw is dependent because one card (Death) has already been drawn, leaving the deck with 77 cards. If a The Tower, Judgement, or The Devil card does turn up after you have already drawn Death, definitely get professional help. Do NOT use a Ouija Board to summons help.
What is the probability that you will draw Death followed by Judgement from a complete deck on sequential draws?
- Probability = (Probability of Event A) times (Probability of Event B)
- Probability = (1 / 78) times (1 / 77)
- Probability = 0.02% or 0.0002 or ^{167}/_{1.000,000} or 1 out of 6,006.
If you draw Death and Judgement consecutively, you are toast. Refer to the second example.
Candy Bars, Your son has just returned from trick-or-treating. He inventories his stash and has: 5 Snickers; 6 Hershey’s bars; 4 Pay Days; 5 Kit Kats; 3 Butterfingers; 2 Charleston Chews; 5 Tootsie Roll bars; a box of raisins; and an apple. You throw away the apple because it’s probably full of razor blades. He throws away the raisins because they’re raisins. After the boy is asleep, you sneak into his room and, without turning on the light, find his stash. Putting your hand quietly into the bag, you realize that it’s too dark to see and all the bars feel alike.
What’s the probability that you’ll pull a Snickers out of the bag?
- Probability = number of favorable events / total number of events
- Probability = 5 / 30
- Probability = 17% or 0.167 or ^{1}/_{6} or 1 out of 6.
This is a probability calculation for 30 independent disjoint outcomes. The draw is independent because only 1 bar is being drawn.
What’s the probability that you’ll pull out a Snickers on your next attempt if you put back any bar you pull out that isn’t a Snickers?
- Probability = number of favorable events / total number of events
- Probability = 5 / 30
- Probability = 17% or 0.167 or ^{1}/_{6} or 1 out of 6.
This is called probability with replacement because by returning the non-Snickers bars to the bag, you are restoring the original total number of bars. The outcomes are independent of each other.
How many bars do you have to pull out before you have at least a 50% probability of getting a Snickers if you put the bars you pull out that aren’t Snickers into a separate pile (not back into the bag)?
- Probability = number of favorable events / total number of events
- 1^{st} bar pulled Snickers probability = 5 / 30 = 17%
- 2^{nd} bar pulled Snickers probability = 5 / 29 = 17%
- 3^{rd} bar pulled Snickers probability = 5 / 28 = 18%
- 4^{th} bar pulled Snickers probability = 5 / 27 = 19%
- 5^{th} bar pulled Snickers probability = 5 / 26 = 19%
- 6^{th} bar pulled Snickers probability = 5 / 25 = 20%
- 7^{th} bar pulled Snickers probability = 5 / 24 = 21%
- 8^{th} bar pulled Snickers probability = 5 / 23 = 22%
- 9^{th} bar pulled Snickers probability = 5 / 22 = 23%
- 10^{th} bar pulled Snickers probability = 5 / 21 = 24%
- 11^{th} bar pulled Snickers probability = 5 / 20 = 25%
- 12^{th} bar pulled Snickers probability = 5 / 19 = 26%
- 13^{h} bar pulled Snickers probability = 5 / 18 = 28%
- 14^{th} bar pulled Snickers probability = 5 / 17 = 29%
- 15^{th} bar pulled Snickers probability = 5 / 16 = 31%
- 16^{th} bar pulled Snickers probability = 5 / 15 = 33%
- 17^{th} bar pulled Snickers probability = 5 / 14 = 36%
- 18^{th} bar pulled Snickers probability = 5 / 13 = 38%
- 19^{th} bar pulled Snickers probability = 5 / 12 = 42%
- 20^{th} bar pulled Snickers probability = 5 / 11 = 45%
- 21^{th} bar pulled Snickers probability = 5 / 10 = 50%
- 22^{th} bar pulled Snickers probability = 5 / 9 = 56%
- 23^{th} bar pulled Snickers probability = 5 / 8 = 63%
- 24^{th} bar pulled Snickers probability = 5 / 7 = 71%
- 25^{th} bar pulled Snickers probability = 5 / 6 = 83%
- 26^{th} bar pulled Snickers probability = 5 / 5 = 100%
This is called probability without replacement. The outcomes are dependent on how many bars have already been taken out of the bag. You would have to try 21 times until you get to 50% probability. 11 Tries will get you to 25% probability. Still, if you returned the bars to the bag you would never have better than a 17% chance of grabbing a Snickers.
Say you picked a Snickers on your first grab. What’s the probability that you’ll pull out a Snickers on subsequent grabs?
- Probability = number of favorable events / total number of events
- 1^{st} bar pulled is a Snickers
- 2^{nd} bar pulled Snickers probability = 4 / 29 = 14%
- 3^{rd} bar pulled Snickers probability = 4 / 28 = 14%
- 4^{th} bar pulled Snickers probability = 4 / 27 = 15%
- 5^{th} bar pulled Snickers probability = 4 / 26 = 15%
- 6^{th} bar pulled Snickers probability = 4 / 25 = 16%
- 7^{th} bar pulled Snickers probability = 4 / 24 = 17%
- 8^{th} bar pulled Snickers probability = 4 / 23 = 17%
- 9^{th} bar pulled Snickers probability = 4 / 22 = 18%
- 10^{th} bar pulled Snickers probability = 4 / 21 = 19%
- 11^{th} bar pulled Snickers probability = 4 / 20 = 20%
Once you do grab a Snickers, the probability that you’ll get another goes down because there are fewer Snickers in the bag. So, the lesson is: Don’t Be Greedy!
You are allergic to peanuts. What’s the probability that you’ll pull out a peanut-free bar (i.e., Charleston Chews, Tootsie Rolls, Kit Kats, or Hershey’s bars)?
- Probability = number of favorable events / total number of events
- Probability = 18 / 30
- Probability = 60% or 0.6 or ^{3}/_{5} or 1 out of 1.67.
Watch out for the bars that may have been produced in facilities that also process peanuts.
What’s the probability that your son will notice you raided his stash?
- Probability = 1.0 or 100%.
Are you kidding?
Next, consider “What Are The Odds?”
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