< Probability and statistics definitions < *Conjunction rule*

## What is the conjunction rule?

The conjunction rule is one of the simplest and most basic rules of probability. It tells us that the probability of the intersection of events (their *conjunction*) cannot be greater than the probabilities of the constituent events.

In other words, **P(A∩B) ≤ P(A) and P(A∩B) ≤ P(B).**

For example, let’s say event A is that the traffic is jammed up on a workday morning (30% probability) and event B is that you are late for work on a weekday morning (20% probability). The conjunction of these events is that there’s bad traffic you’re late for work. Assume:

- The two events are dependent, because bad traffic can make you late, and
- You are 50% more likely to be late if there is traffic.

Since being 50% more likely to be late means increasing the original probability of 20% by 50%, the adjusted probability of being late given bad traffic is

**P(B|A) = 0.2 x 1.5 = 0.3 or 30%**

Then the probability of hitting bad traffic and being late is

**P(A and B) = P(A) x P(B|A) = 0.3 x 0.3 = 0.09 or 9%.**

## The conjunction fallacy

The **conjunction fallacy** happens when people mistakenly assume that the probability of the conjunction can be higher than individual probabilities. In our traffic and work example, a person might incorrectly calculate the probabilities as follows:

They correctly know:

- The probability of hitting bad traffic is 30%: P(A) = 0.3,
- The probability of being late is 20%: P(B)= 0.2.

A person committing the conjunction fallacy might say something like:

“OK, if there’s a 30% chance of traffic and a 20% chance of being late, then the chance of both happening together is probably even higher, maybe 40% or more, because if there’s traffic, I’ll definitely be late!”

## What is the restricted conjunction rule?

The restricted **conjunction rule** states that the probability of two events happening at the same time equals to the product of their individual probabilities. The restricted rule only applies when the two events are independent.

For example, if winning a work raffle today has a 0.2 probability and finding a penny on the ground today has a 0.5 probability, the probability of both events happening at the same time is:

** 0.2 x 0.5 = 0.1.**

In other words, your probability of winning that work raffle today and also finding a penny on the ground is 0.1 or 10%.

As another example, let’s say we were looking into the relationship between smoking and lung cancer. The conjunction rule can be applied to calculate the probability a person will be a smoker and develop lung cancer. If the probability of smoking is 20% and the probability of developing lung cancer is 10%, the probability of a smoker getting lung cancer would be

20% x 10% = 2%.

Note that we can use percentages or decimals, but it’s more traditional in probability to use decimals because it aligns with the theory of probability where 0 represents an impossible event and 1 represent a certain one. Decimals are also easier to multiply, divide and perform other mathematical operations.

## Conjunction vs multiplication rule

Rule | Description | Example | Value |
---|---|---|---|

Conjunction rule | Calculates the probability of two events occurring at the same time | Probability of flipping a coin and getting heads and tails |
0% |

Multiplication rule | Calculates the probability of two events occurring independently | Probability of flipping a coin and getting heads or tails |
50% |

*Table of differences between the conjunction and multiplication rule.*

The conjunction rule and the multiplication rule are both used in calculating the probability of two events happening at the same time, but they differ in their applications.

- With the
**conjunction rule,**both events must happen at the same time. For example, the probability getting both heads*and*tails on a coin flip is 0, because it’s impossible for both outcomes — heads and tails with the same coin — to happen at the same time. - On the other hand, with the
**multiplication rule**, the events can happen at any time, and the occurrence of one event does not impact the occurrence of the other. For example, the probability of flipping a coin and getting either heads or tails is 50%, because each outcome is equally likely on any given flip.

As noted above, the restricted conjunction rule applies only when the two events do not influence each other. For example, the probability of flipping a coin and getting heads, then flipping it again and getting tails, equals the probability of getting heads on the first flip multiplied by the probability of getting tails on the second flip. These events are independent since the outcome of the first flip doesn’t impact the outcome of the second flip.

However, the Conjunction Rule does not hold when the two events are dependent. For example, the probability of flipping a coin and getting heads both times isn’t equal to the product of the probabilities of getting heads in each flip, as the outcome of the first flip affects the outcome of the second flip.

## References

- Probability. Retrieved June 23, 2023 from: https://www.nku.edu/~garns/165/ppt9_3.html