Last Updated on 16 September 2023

You will need to know probability in order to properly analyze the data you are collecting in your process improvement project. Probability is generally the study of how frequently an outcome occurs, such as the possibility of getting a ‘3’ when you roll a dice.

An outcome is the result of the experiment, so in this case the number on the dice roll.

The sample space is all the possible outcomes (1, 2, 3, 4, 5 or 6)

## Venn diagrams

You can show probability of various outcomes using a Venn diagram. You may remember these from school; they’re the sometimes overlapping circles that group different outcomes. If you draw all the possible outcomes on a Venn diagram, an event is drawn a a circle round all the outcomes that would provide that event. An event of an odd roll of the dice would be a circle round 1, 3 and 5.

## Mutual exclusivity

Mutually exclusive events are those where the two events can’t happen at the same time. As this would mean the two circles would have no outcomes in common, you can easily see these on the Venn diagram from those circles with no overlap. Going back to the dice, a ‘roll of 4’ and an ‘even dice roll’ are mutually exclusive, as they would have no outcomes in common.

## Independent events

A key factor in probability is the interdependence of the events. This is if an event occurs, does it make it more or less likely for the event to reoccur. A dice roll of 3 doesn’t make the next roll more or less likely to be a 3, so the consecutive rolls are independent.

## Dependent events

Dependent events are those that affect the probability of the next. I live in famously rainy England, so my example is rain related. We all look at rain reports for the day, which give us the e.g. the probability that it will rain between 1pm and 2pm. This is dependent on the rain between midday and 1pm. If it rains in the earlier time, it makes later rain more likely, and no rain in the earlier slot makes later rain less likely.

## Complementary probability

A complementary probability is the probability that something doesn’t happen. The probability of event A occurring is P(A); the complementary probability is written P(A’), where:

P(A’) = 1 – P(A)

## Intersection

Intersection is the probability that both events A and B occur. It is the probability of the area that overlaps between the circles of A and B. In mutually exclusive events, the intersection is zero. How you calculate the probability of intersection of two events depends on their interrelation:

## Multiple events

### Independent events

The probability of a series of independent events occurring is simply the two probabilities multiplied together. It is the intersection of the two events mentioned earlier.

The probability of rolling a 1 on a dice is 1/6, and so the probability of rolling a 1 twice in a row is 1/6 x 1/6 = 1/36. This is the int

### Dependent events

The probability for dependent events is more complicated, as it depends on the relationship between the two events.

The probability is instead Probability of A x Probability of B when A has already occurred. This is written P(A) x P(B|A).

### Union

Union is the probability that either event A or B occurs. This is the probability of the two circles in total in the Venn diagram. The calculation becomes obvious when looking at the diagram. It is the probability of A plus the probability of B less the intersection which otherwise would be counted twice.

## Factorial

A factorial is used in a lot of formulas, such as when dealing with distributions. You will encounter a lot of distributions in Six Sigma, so you’ll find it a lot. Fortunately it’s quite simple.

You’ll see it written as n!, e.g. 5!. To calculate this, you simply multiply the number by all the integers below it until you hit 1, so:

5! = 5 x 4 x 3 x 2 x 1 = 120

In case it’s not clear, 1! = 0! = 1, and you can’t have a factorial for anything except positive integers (whole numbers).

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