Basics of Probability

Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. There are several basic rules of probability that can be used to help determine the probability of multiple events happening together, separately, or sequentially. This article covers the fundamentals of probability, which are important both when conducting or interpreting the results of clinical trials, and when making clinical decisions for patients based on the probability of different outcomes.

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Probability and Random Phenomena

Probability

Probability is a mathematical tool used to study randomness and provide predictions about the likelihood of something happening. 

  • Abbreviations:
    • Probability is abbreviated as P(event).
    • Example: P(A) refers to the probability of event A happening.
  • Types of probability include:
    • Theoretical probability: mathematical models and not observations
    • Relative frequency: observations and/or measurements
    • Personal or subjective probability: nothing other than personal feelings

Example of probability:

If you toss a fair coin, the theoretical probability of heads on any given toss is 50%.

Law of large numbers (LLN)

  • When an event is repeated over and over again, the proportion of times that the event results in a particular outcome will settle around a particular number.
  • That particular number is the probability of the outcome.
  • Refers to long-term outcomes (rather than short-term outcomes)

Example of short-term vs long-term outcomes:

If you toss a fair coin, there is a 50% chance of getting heads on any given toss. If you toss 10 heads in a row, this does not increase the chance of getting heads on the next toss (a short-term outcome). The LLN does mean that over a large number of coin tosses, the frequency of getting heads will be close to 50%.

Probability assignment

  • If the probability of an event = 0, the event will never occur.
  • If the probability of an event = 1, the event will always occur.
  • The probability of an event happening can be distributed between the possible outcomes associated with it.
  • The probability of all outcomes adds up to 1.

The probability of an event (A) happening is between 0 (absolute certainty of it not occurring) and 1 (absolute certainty of it occurring):

$$ 0\leq P(A)\leq 1 $$

Summation of all the probabilities of all possible outcomes in a sample space add up to 1:

$$ P(S) = 1 $$

Random phenomena

Random phenomena are situations where the possible outcomes are known, but the one that will happen is unknown.

  • They can be observed during what are known as trials, where outcomes occur.
  • Sample space: the set of all possible outcomes

Basic Rules of Probability

The complement rule

Some events only have 2 possible outcomes: event A occurs, or event A does not occur. The complement of event A is that event A does not occur, and is represented as AC. The probability of AC occurring equals 1 minus the probability of the event itself (A).

$$ P(A^{C}) = 1 – P(A) $$
A venn diagram illustrating the complement rule

A Venn diagram illustrating the complement rule:
The entire gray box represents the sample space, which equals 1. Event A represents 1 portion of the box, and event A not occurring represents the remaining portion of the sample space. From the “Introduction to Probability” lecture

Image by Lecturio. License: CC BY-NC-SA 4.0

Example: You have a 1-in-4 chance of drawing a club from a standard deck of cards. What is the probability that you will not draw a club?

Answer: In this example, drawing a club is event A and not drawing a club is AC. If the chance of drawing a club is 0.25, then AC = 1 ‒ 0.25, which is 0.75.

Rule of disjoint events

If 2 or more events cannot occur simultaneously, they are called mutually exclusive or disjoint events. While it is not possible for the 2 disjoint events to occur simultaneously, it is possible for neither of them to occur. 

A venn diagram illustrates the rule of disjoint events

A Venn diagram illustrating the rule of disjoint events:
The area of the box represents the entire sample space, which equals 1. Circle A represents the probability of event A occurring, and circle B represents the probability of event B occurring.
The circles do not overlap, indicating they are mutually exclusive and cannot occur simultaneously. It is possible, however, for neither event to occur. Notice how they do not overlap.

Image by Lecturio. License: CC BY-NC-SA 4.0

When 2 events (A and B) are mutually exclusive (or disjoint), the probability that A or B will occur is the sum of the probability of each event.

$$ P(A\cup B) = P(A) + P(B) $$

This rule can be applied to any number of disjoint events. For instance, to find the probability of either A, B, or C occurring, you can simply add P(A) + P(B) + P(C), assuming all 3 are completely mutually exclusive events.

Example 1: 

  • You come to a stop light. There is a 35% chance you will pull up to the light while it is green, a 5% chance you will pull up to the light while it is yellow, and a 60% chance you will pull up to the light while it is red. When you pull up to a stop light, what is the chance that the light will be green or yellow? 
  • Answer: Since the light cannot be green and yellow at the same time in this example, there are only 3 possibilities: P(green) can simply be added to P(yellow), which is 0.35 + 0.05 = 0.4. Thus, the chance that the light will be either green or yellow is 40%.

Example 2: 

  • You have a standard deck of cards. What is the probability of drawing either a club, spade, or heart?
  • Answer: The card you draw will only have 1 of the 4 suits at a time; thus, these are all mutually exclusive events. Therefore, P(club) + P(spade) + P(heart) = 0.25 + 0.25 + 0.25 = 0.75. There is a 75% chance you will draw a club, spade, or heart.

The multiplication rule for independent events

Events are independent when the probability of one does not affect that of the other (Note: Disjoint events cannot be independent events (Example 2)). The probability of 2 independent events both occurring equals the product of the probabilities of events A and B.

$$ P(A\cap B) = P(A)P(B) $$

Example 1: 

  • Event A is the probability of drawing a club from a deck of cards, which is 13 / 52, or 0.25. Event B is the probability of drawing a face card, which is 12 / 52, or 0.23. What is the probability of drawing a club that is also a face card?
  • Answer: These 2 events are independent of each other (the probability of drawing a club has no effect on the probability of drawing a face card); thus, the probabilities can simply be multiplied together. 0.25 x 0.23 = 0.057 or 5.7% (which is 3 / 52, representing the King, Queen, and Jack of clubs).

Example 2: Disjoint events cannot be independent of each other

  • Disjoint events are 2 events that cannot happen at the same time: e.g, a stop light cannot be both red and green at the same time. If the light is green, it cannot also be red.
  • Independent events can happen simultaneously: e.g., a card can be both a club and a face card.

General addition rule

The probability of an event (A), or another event (B), or both happening is given by the equation: 

$$ P(A\cup B) = P(A) + P(B) – P(A\cap B) $$
A venn diagram illustrating the general addition rule of probability

A Venn diagram illustrating the general addition rule of probability:
The area of the entire gray box represents the entire sample space, which equals 1. There is a circle that encapsulates the probability of event A occurring (green), and another for event B (red). The area where they overlap represents the probability of both events occurring simultaneously.
Therefore, if you were to simply add the area of the green circle to that of the red circle, the overlapped area would be counted twice. Visually, the total probability of either A, B, or both occurring can be represented as P(A) + [P(B) ‒ P(A&B)].

Image by Lecturio. License: CC BY-NC-SA 4.0

Example: You have a stack of money with 4 bill denominations: $1, $5, $10, and $20. Event A represents drawing an odd-numbered bill; event B represents drawing a bill between $4 and $12. What is the probability of A or B happening?

Answer: Note that $5 is in both events; therefore, they are not disjoint or mutually exclusive. Thus, we cannot simply add P(A) + P(B), because we will have counted the probability of a $5 bill being drawn (P($5)) twice. We, therefore, must subtract P($5) so that it is only counted once in the end. If the chances of drawing each bill are the same, then the probability of drawing each individual bill is 1 in 4, or 25%.

So, to answer our question, 1st we can calculate P(A), which equals P($1) + P($5) = 0.25 + 0.25 = 0.5. Similarly, P(B) = P($5) + P($10) = 0.25 + 0.25 = 0.5. We know that P($5) by itself is 0.25. So, overall, 0.5 + 0.5 ‒ 0.25 = 0.75, which represents 3 out of the 4 bills (the $1, $5, and $10 bills, which are all included in either event A or B).

Pitfalls

  • Beware of probabilities that do not add up to 1.
  • Do not add probabilities of events if they are not disjoint.
  • Do not multiply probabilities of events if they are not independent.
  • Disjoint events cannot be independent.
  • Do not use LLN to describe short-term events.
  • Consider whether assuming events to be independent is reasonable.

Conditional Probability

  • Conditional probability of event B is the probability that B will occur knowing that event A has already occurred.
  • Notated as P(B|A)
  • Conditional probability of independent events is simply the probability of event B, meaning P(B|A) = P(B).
    • Example: A person wishes to draw 2 clubs in a row from a standard deck. Assuming their 1st card is a club; what is the probability that the 2nd card will also be a club?
    • Answer: Each draw is independent of the last draw, so the conditional probability, in this case, can be described as: P(Drawing a 2nd club|1st card is clubs). As there are 13 cards in each suit, and 1 has already been drawn (“event A”), this leaves 12 clubs out of a total of 51 cards. Therefore, the answer is 12 / 51 or 23.5%.
  • Conditional probability of events that are not independent represents the probability of both events occurring, meaning P(B|A) = P(A)*P(B).
    • Example: A student applying to college has an 80% chance of being accepted. On-campus housing is available for 60% of accepted students. What is the chance of acceptance and getting on-campus housing?
    • Answer: 0.8 x 0.6 = 0.48 or 48%
  • Conditional probability of more than 2 events requires consideration of all preceding events.
    • Example: The same student above knows that of students who get on-campus housing, 90% have at least 1 roommate. What is the chance this student gets accepted, gets on-campus housing, and has at least 1 roommate?
    • Answer: 0.8 x 0.6 x 0.9 = 0.432 or 43.2%

References

  1. Haidich, A.B. (2010). Meta-analysis in medical research. Hippokratia, 14 (Suppl 1): pp. 29–37.
  2. Smith, V., Devane, D., Begley, C.M., Clarke, M. (2011). Methodology in conducting a systematic review of systematic reviews of healthcare interventions. BMC Medical Research Methodology, 11 (1).
  3. Rind, D. (2019). Proof, p-values, and hypothesis testing. UpToDate. Retrieved May 25, 2021, from https://www.uptodate.com/contents/proof-p-values-and-hypothesis-testing
  4. Mahutte, N., Duleba, A. (2021). Evaluating diagnostic tests Diagnostic tests Diagnostic tests are important aspects in making a diagnosis. Some of the most important epidemiological values of diagnostic tests include sensitivity and specificity, false positives and false negatives, positive and negative predictive values, likelihood ratios, and pre-test and post-test probabilities. Epidemiological Values of Diagnostic Tests. UpToDate. Retreived May 25, 2021, from https://www.uptodate.com/contents/evaluating-diagnostic-tests

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