Statistics: Probability

Determine the Likelihood of an Event

Similar to my Distribution articles, Probability is another concept which focuses on how likely an event will occur.

Single Event Probability

The following formula returns the likelihood of an event using how many events qualify for the desired outcome (criteria), divided by the total number of possibilities.

To illustrate this formula, I will begin with the following data set to determine the probability the student Chris Winns will receive an ‘A’ grade.

First, I will populate the numerator (match criteria). Since the only outcome which matches our criteria is the grade ‘A’, that leaves only 1 for criteria.

Next, I will populate the denominator with the total number of possible outcomes. Since there are only four letter grades (F, D, C, B, A).

By dividing the numerator by denominator, I see there is a 20% probability of Chris Winns receiving an “A” grade.

As you can see in the following table, a 20% probability means this event is less likely to occur.

Probability Likelihood
0 0% chance
1 100% chance
0.5-.99 More likely to occur
0-0.49 Less likely to occur
.05 Equal chance

Note: In this example, the possibilities involved which match our criteria don’t include duplication. For example, if this example involved flipping two coins and the matching criteria was seeing the heads side, instead of the number being 2 (two coins each with a matching criteria of 1 side of the coin), the matching criteria remains only 1 to avoid duplicating the count of possible matching criteria.

Addition Rule: Avoiding Duplication When Determining Matching Criteria in Probability

When counting possible outcomes which match criteria, use this formula:
Probability(Event1 or Event2) = Probability(Event1) + Probability(Event2) – Probability(Event1 and Event2)
In the coin example this formula would translate to:
Probability(Event1 or Event2) = 1 (heads of first coin) + 1 (heads of second coin) – 1 (number of times where both might occur)
Probability(Event1 or Event2) = 2 – 1
Probability(Event1 or Event2) = 1
This match criteria of 1 ensures the possibility of the coins turning heads isn’t counted more than once.

This example reflects determining probability for mutually exclusive events in that the first coin toss in no way affects the second.

The formula may also be rewritten more briefly as:
P(E1 ∪ E2) = P(E1) + P(E2) − P(E1 ∩ E2)
E denotes an event.
∪ denotes “and” and means a “union” of either E1 or E2 or both.
∩ denotes “or” and means an intersection of both E1 or E2.

Multiplication Rule: Determining Probability for Multiple Events

Using two six-sided dice as an example, I will use the same formula as above to determine the likelihood of rolling a 1 for each dice when rolled at the same time.
In this formula however, I will multiple the probabilities.

Now, I’ll enter my match criteria (1 if one change) and possibilities (6 because there are 6 different possible values). Also, since the occurrence of the first roll has no effect on the occurrence of the second roll so the second probability is not changed.

I can see the probability of rolling two six-sided dice and each dice landing with 1 is 1 out of 36 chances.

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