How to calculate probability

Marginal probability: the probability of an event occurring (p(A)), it may be thought of as an unconditional probability.  It is not conditioned on another event.  Example:  the probability that a card drawn is red (p(red) = 0.5).  Another example:  the probability that a card drawn is a 4  (p(four)=1/13).

Joint probability:  p(A and B).  The probability of event A and event B occurring.  It is the probability of the intersection of two or more events.  The probability of the intersection of A and B may be written p(A ∩ B). Example:  the probability that a card is a four and red =p(four and red) = 2/52=1/26.  (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds).

Conditional probability:  p(A|B) is the probability of event A occurring, given that event B occurs. Example:  given that you drew a red card, what’s the probability that it’s a four (p(four|red))=2/26=1/13.  So out of the 26 red cards (given a red card), there are two fours so 2/26=1/13.


Probability of Two Events Occurring Together: Overview

probability of two events happening together

Answering probability questions can seem tricky, but they all really boil down to two things:

  1. Figuring out if you multiply or add probabilities.
  2. Figuring out if your events are dependent (one event has an influence on the other) or independent (they have no effect on each other)

Should I multiply or add probabilities?

You would add probabilities if you want to find out if one event or another could happen. For example, if you roll a die, and you wanted to know the probability of rolling a 1 or a 6, then you would add the probabilities:Probability of rolling a 1: 1/6Probability of rolling a 6: 1/6So the probability of rolling a 1 or a 6 is 1/6 + 1/6 = 2/6 = 1/3.

The probability of both events happening together on the same die is zero, at least with a single throw. But if you wanted to know the probability of rolling a 1 and then rolling a 6, that’s when you would multiply (the probability would be 1/6 * 1/6 = 1/36).

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Learning when to add or multiply can get really confusing! The best way to learn when to add and when to multiply is to work out as many probability problems as you can. But, in general:If you have “or” in the wording, add the probabilities.If you have “and” in the wording, multiply the probabilities.This is just a general rule — there will be exceptions!

Dependent vs. Independent

Dependent events are connected to each other. For example:

  • In order to win at Monopoly, you have to play the game
  • In order to find a parking space, you have to drive
  • Choosing two cards from a standard deck without replacement (the first choice has a 1/52 chance, the second a 1/51).

Independent events aren’t connected; the probability of one happening has no effect on the other. For example:

  • playing Monopoly isn’t connected to winning at Scrabble
  • Winning the lottery isn’t connected to you winning at Monopoly
  • Choosing a card and then rolling a die are not connected

If you aren’t sure about the difference between independent and dependent events, you may want to read this article first:Dependent or Independent event? how to Tell.

Tip: Look for key phrases in the question that tell you if an event is dependent or not. For example, when you are trying to figure out the probability of two events occurring together and the phrase “Out of this group” or “Of this group…” is included, that tells you the events are dependent.

Probability of Two Events Occurring Together: Dependent

The equation you use is slightly different.P(A and B) = P(A) • P(B|A)

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where P(B|A) just means “the probability of B, once A has happened)

Sample problem: Eight five % of employees have health insurance. Out of those 85%, 45% had deductibles higher than $1,000. What percentage of people had deductibles higher than $1,000?”

Step 1: Convert your percentages of the two events to decimals. In the above example:

85% = .85.45% = .45.

Step 2: Multiply the decimals from step 1 together:

.85 x .45 = .3825 or 38.35 percent.

The probability of someone having a deductible of over $1,000 is 38.35%

That’s how to find the probability of two events occurring together!

Tip: Sometimes it can help to make a sketch or drawing of the problem to visualize what you are trying to do. The following diagram shows the group of people (85% of the population) and the subgroup (45% of the population), making it more obvious that you should be multiplying (because when you translate 45% of the 85% (have insurance with high deductibles) to math, you get .45 * .85).probability of events happening together 2


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How to Manipulate among Joint, Conditional and Marginal Probabilities

The equation below is a means to manipulate among joint, conditional and marginal probabilities.  As you can see in the equation, the conditional probability of A given B is equal to the joint probability of A and B divided by the marginal of B.  Let’s use our card example to illustrate.  We know that the conditional probability of a four, given a red card equals 2/26 or 1/13.  This should be equivalent to the joint probability of a red and four (2/52 or 1/26) divided by the marginal P(red) = 1/2.  And low and behold, it works!  As 1/13 = 1/26 divided by 1/2.  For the diagnostic exam, you should be able to manipulate among joint, marginal and conditional probabilities.

Bayes’ Theorem

Bayes’ theorem: an equation that allows us to manipulate conditional probabilities. For two events, A and B, Bayes’ theorem lets us to go from p(B|A) to p(A|B) if we know themarginal probabilities of the outcomes of A and the probability of B, given the outcomes of A. Here is the equation for Bayes’ theorem for two events with two possible outcome (A and not A).

Screenshot 2014-08-24 22.37.48

Let’s assume we know that 1% of women over the age of 40 have breast cancer. 


Let’s assume that 90% of women who have breast cancer will testpositive for breast cancer in a mammogram.

[p(positive test|cancer)=0.9]

Eight percent ofwomen that do NOT have cancer will also test positive.

[p(positive test|no cancer)=0.08]

What is the probability that a woman has cancer if she tests positive [p(cancer|positive test)]?

We will call p(cancer) = P(A), and the P(positive test) = P(B). We want to know P(A|B)–the probability of having cancer if you have a positive test.

Screenshot 2014-08-24 22.46.21

Using Bayes’ theorem, we calculate that the likelihood that a woman has breast cancer, given a positive test equals approximately 0.10. This makes intuitive sense as (1) this result is greater than 1% (the percent of breast cancer in the general public).

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