## Conditional Probability Calculations… Continued

Let’s roll two dice to illustrate the point. Each roll will show a number from {1,…6} with equal probabilities of 1/6.

The goal is to roll a **sum** total value of 9, 10, 11 or 12. This can be achieved in 10 of 36 possible outcomes: “P(*S=9-12*) = 10/36 = 0.2778″.

Although each dice roll is independent, the probability of the sum from those rolls does depend on each.

Let’s say that “*D1=4*” is the outcome inwhich the first dice roll shows a four. P(*D1=4*)= 1/6. From the image, we see that the intersection of these events, “D1=4” and “S=9-12”, only has two outcomes. Therefore P(*D1=4*)∩(*S=9-12*) = 2/36.

The conditional probability of the total value being 9-12, given that the first dice roll yields a 4, is “P(*S=9-12*|*D1=4*) = P(*D1=4*∩*S=9-12*) / P(*D1=4*) = (2/36) / (1/6) = (2/36) * (6/1) = 1/3″.

However, the conditional probability that the first dice roll is a 4, given that the total sum is known to be 9-12, is “P(*S=9-12*|*D1=4*) = P(*D1=4*∩*S=9-12*) / P(*S=9-12*) = (2/36) / (10/36) = (2/36) * (36/10) = 2/10 = 1/5″.

## Common Problems With Conditional Probability Calculations

Calculating conditional probability can be a powerful technique. There are many situations where we have partial knowledge of an event, but still need to know the probability of the final outcome. A game of poker with repeated rounds of betting represents such a situation.

On the other hand, it can be useful to know how likely one input was, considering that the final outcome is known. If the car won’t start on a cold winter’s morning, is it more likely that the battery failed or that a thief siphoned the gasoline? If the gas tank is nearly empty, is it more likely that a thief or a forgetful driver is responsible?

The important trick for computing conditional probability is to understand what is the “given”, versus what is still unknown.

**Reference**:

Weisstein, Eric W. Conditional Probability. MathWorld-A Wolfram Web Resource. (1999-2011). Accessed January 17, 2012.

**Click to Return to Page One: Pitfalls in Conditional Probability**

Decoding Science. One article at a time.