# Now I Know My p(A), p(B), p(C) …

This is not a primer or a refresher on probability but let us start with some probability calculations anyway,

1. What is the probability of getting Heads when you toss a fair coin?
2. How about getting  8 when you roll a pair of fair dice?
3. I have a rod that is 1 meter long. I draw two random numbers between 0 and 1 and cut the rods into three parts using the random numbers as lengths of pieces. What is the probability the three pieces can be arranged as a triangle?

Some are easy, some are computationally intensive but eventually we can get to the answer. We arrive at the answer by counting – we count all possible outcomes (Sample Space N) and count the number of desired result ( n) and find the probability a n/N.  (I think the answer to 3 is 0.25)

This is the traditional (also known as frequentist approach (because it involves counting frequencies of events) to probability.

But consider these,

1. You see the techcrunch headline about a new startup. Just from that headline, what is the probability that the startup will be a big success?
2. You see a salon on Main St, what is the probability that the next customer who walks in will be named Jonathan?

These are not easy to answer – definitely not by counting the sample space or the outcomes. Probability in these cases ceases to be a ratio of two countable events and becomes a representation of hunch, degree of belief, gut feel or a hypothesis.

I said it, hunch, belief, gut feel …

Probability now becomes a measure of our certainty (or uncertainty) about the outcome.  We are in the realm of  Bayesian probability. Despite the esoteric name, we all practice this in our every day reasoning.

A hunch, belief or hypothesis  has to start somewhere.

In some cases we start with  a probability value that is no different from the traditional approach. We remember reading somewhere that 90% of startups do not go big. So we answer the startup question as, 10%.   Similarly, the answer to the freemium question is 3%.

In some cases, it occurs from our domain knowledge and years of experience working with a specific area.

In some cases, it just occurs to us and we “just feel it in our gut”. So we state that the probability of next customer being Jonathan as 17%.

A hunch, gut feel or belief is not all bad when that is all we can make.

But problems arise when we trust our guts or pay grades,  even in the presence of prior knowledge or fail to take new knowledge into account to improve our hunch.

1. We fail to recognize that our initial assessment of the probability is based on limited observations.  One cannot make claims about safety of nuclear reactors for the next 500 years based only on 60 years of observation.
2. We straddle unknowingly from probability being a degree of our belief to the traditional definition of ratio of observed events. If the belief /hunch itself is based only on limited knowledge and information then we stand to commit serious errors of decision making, confusing our belief with real likelihood of occurrence of events.
3. Since we tend to overstate our knowledge and suffer from optimism bias  we overestimate the probability of favorable events and underestimate the probability of less favorable events. We then confidently state our degree of belief as actual measure of likelihood of occurrence of an event.
4. We stick to our initial estimate even when we uncover new information. In the traditional approach to probability, the values do not change.
p(H) = p(T) = 0.5  for any fair coin
But in the Bayesian world, probabilities change when we learn new information.
For example, if you learn that the name of the salon is Rapunzel, you will restate the probability of next customer being Jonathan as a very low number.

Knowing p(A), p(B), p(C)  is not any more as simple as A, B, C.

Next time we state a probability value, let us stop and ask whether we are stating our belief or the real likelihood of an event.

Notes:

To the question of finding the probability of a free user upgrading to premium version, see here.

The answer to the triangle question is based on the sides rule for triangles.

For discussion of gut vs. mind see here.

For the need for evidence based marketing, see here.

## 6 thoughts on “Now I Know My p(A), p(B), p(C) …”

1. I have just blogged on the difference between uncertainty and probability at http://djmarsay.wordpress.com/2011/06/07/which-mathematics-of-uncertainty-for-todays-challenges/. I think that our concerns overlap.

For example, suppose that you are to toss a coin for something that you value. You have a coin that you have used frequently on similar occasions and found to be fair. Your opponent asks you to guess the probability of heads of a coin that he has. It seems to me that you have to guess 0.5, in which case if he offers you a small amount if you agree to use his coin then ‘rationally’ you should agree. But I would much prefer my own coin, about which I had much less uncertainty. So for both of us, I think, probability is sometimes only a part of uncertainty, and perhaps not the most important part.

Like