Dice Paradox and Conditional Probability

Let's look at another example.

The probability of rolling a 7 with 2 dice is 1/6. We can verify this by first listing all possible combinations of the 2 dice:

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).

Now, we scan through this list of 36 possible events and count all events in which the sum of the 2 dice is 7. The number of events in which this is true is 6. Check for yourself. THus the probability of obtaining a 7 is 6/36 or 1/6. This is true regardless of whether you throw one die followed by the second die or throw them both at the same time.

If I bet on 7 for any random throw of the dice, I would have a chance of winning 1/6 of the time. Now, being an astute gambler, I would like to improve the odds of me winning. Let's have an impartial observer take a look at the dice after they are rolled, but not allow me to see the result. Let's say our favorite number is '4', so anytime the observer sees a '4' on at least one of the die thrown, he shouts out "I see a four!" With this extra bit of information provided before we make our bet, we have improved our odds of winning from 1 in 6 to 2 in 11! We have improved our odds in spite of the fact that we have not changed the probability of the resulting roll of the dice.

How can that be? Well, we just need to go back to our list of all possible events and count the number of events in which a '4' appears on at least one of the die. 11 of the 36 combinations have at least one '4' appear. Now of those 11, how many have a sum equal to 7? Exactly 2. Since we consider only consider betting on those dice rolls in which the number '4' is called out, we have only 11 possibilities and of those only 2 can sum up to 7.

This is an example of conditional probability. This extra bit of information we receive can affect the original probability. The conditional probability theorem gives us a shortcut method to obtain the probability without having to count all combinations and subset of combinations as we did in this example. We will explore the formulation of this conditional probability equation next time.

To be continued...