Conditional Probability is Not Commutative: Formulas and Examples

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Conditional Probability Given the First Dice Roll: Image by Mike DeHaan

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.

Conditional Probability Given the Sum: Image by Mike DeHaan

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=4S=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=4S=9-12) / P(S=9-12) = (2/36) / (10/36) = (2/36) * (36/10) =  2/10 = 1/5″.

Common Problems With Conditional Probability Calculations

Poker illustrates a real-life use of probability calculations: Photo by Images_of_Money

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.

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