A Game Of Dice Continued

Recall from our previous example we obtained the impossible probability of rolling a single die a number of times and obtaining a value greater than one. To help understand the flaw in our reasoning let's make this even simpler. What is the probability of rolling at least one '4' after 2 tosses of a single die?

Intuitively we might say 1/6 probability of rolling a '4' on the first toss plus 1/6 probability of rolling a '4' on the second toss gives us 1/3. So is 1/3 the right answer?

What about the case where we roll a '4' on the first toss and a '4' on the second toss?

If we make a list of all possible combinations of the value of the first roll of the die and the value of the second roll of the die, we will end up with with 36 combinations:
(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), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).

Recall our original question: What is the probability of rolling at least one '4' after two rolls of the die?

So all we have to do is to count the number of times we see at least one '4' show up in the 36 possible combinations. From the list we see eleven instances in which at least one '4' is present.

To find the probability we take the ratio of the number of combinations in which at least one '4' is present and the total possible combinations we can have.

Our answer is 11/36.

We have an 11/36 probability of rolling at least one '4' after 2 tosses of a die.

Mysterious? Counter intuitive. Absolutely! But with some straight forward logical reasoning we have come up with the true probability.

To be continued...

A Taste of Monte Carlo Simulation

There are many areas in which we want to predict the future.

For example what will happen to my retirement account in five years? What will the weather be like next week? What is the inflation rate going to be over the next few years? How likely is it that my iPhone will fail or break within a year?

In traditional forecasts, we create a model that projects outcomes based on certain inputs or variables. These are entered into a model which relates the inputs mathematically to produce outputs. These inputs are single-point estimates or our "best guess." Since the inputs are point estimates, the outputs will be single point estimates as well. In other words, we have to have complete confidence in the accuracy of our inputs (e.g. inflation rate, failure rates, investment rate of return, GDP, etc.) if we want to believe the output is exactly correct.

Since forecast inputs are effectively predicting the future, for any real world phenomena the actual values are not known with absolute certainty.

To incorporate this uncertainty, each single-point input estimate can be replaced by a probability distribution that more accurately reflects the range of possibilities for that input.

The output will then be a range of possibilities or a probability distribution. To obtain this probability distribution on the output, we sample the input. The input is sampled based on the probability distribution associated with this input. For each sample we obtain one output value.

Using Monte Carlo simulation we repeat this process over and over multiple times to get a range of outputs based on a range of inputs.
This range of outputs represents the output probability distribution which gives us a more realistic set of possibilities for the future.

With this information we can begin to answer questions such as:
How likely is it that I will achieve millionaire status when I retire?
How likely is it that 100 of these units will fail within the next 10 years? what is the most important contributing factor to sales growth?

A Game of Dice

Let’s play a game. Take one die and roll it. Remember the value of the roll. Then roll it again. What is the probability that after two rolls ‘4’ appears at least once? Ok, we reason, for the first roll of the die we have a 1/6 chance of a ‘4’ since the die has 6 possible values. And since the roll of the die the second time is not influenced by the value of the die we obtained after the first roll, we assign a 1/6 chance of a ‘4’ for the second roll as well. Now since we have 2 chances of getting a ‘4’ our odds should double then. So we add the two probabilities together. 1/6 + 1/6 = 1/3. We expect a 1/3 probability of seeing at least one ‘4’ after rolling the die twice. Now I ask “What is the probability that after 10 rolls of the die, a ‘4’ appears at least once. Ok, simple, let’s add them up. 1/6 probability for each roll of the die times the number of times we roll the die, i.e. 10 should give us the number. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + /1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 10/6. “10/6 probability of getting a ‘4’”, we proudly say.

Wait a minute! How come you have a probability that exceeds 1? I thought a probability value can only range from 0 (absolutely impossible) to 1 (happens always every time). Obtaining 10/6 or 167% must mean that we are absolutely certain this will happen after 10 rolls and just to make sure we’ve added 67% padding on top for good measure!

Well, something must be wrong in our calculations or thinking. We know a probability of any event or series of events can never exceed 1. Where did we go wrong?

Politics In Organizations

Why is it that so many of us are dissapointed with the performance of politicians and wall street bankers? Some answers may be found in the new book "The Dictator's Handbook."