We do not like uncertainty. Most of us have a hard time understanding probability in an intuitive way. For example, it is hard to imagine ourselves in Hawaii with 50% probability and in Egypt with 50%. We either think of lying on the beach or watching the pyramids. Superimposing one on top of another is not possible for us.

This is, in some sense, quite problematic for us in terms of investments. Investments have a way of depending on chance. In separate situations we will end up with different returns.

Most investors vacillate between extreme pessimism and extreme optimism about the stocks they own. When the market is crashing, we can only see the downside and when it is recovering then we see only the upside. **Probability**: If you roll a dice, you might end up with 1, 2, 3, 4, 5 or 6. If the dice is symmetric i.e., fair then there are 6 different outcomes and each one of them is equally likely. So, out of these 6 outcomes, there is exactly one which ends up with 4. In mathematics we say, “The probability of getting 4 when you roll a fair dice is 1/6 or 0.16666...” The probability is always a number between 0 and 1. If the probability is 1 then we say that the event will definitely happen. There is no uncertainty. If the probability is 1/2 then the event has equal chance of happening and not happening. If the probability is 0 then the event will definitely not happen. Here is what Buffett has to say about probabilities and investment success.

If you think you understand probabilities now, I give you a few brain teasers. You can write the answers in the comment section. Do not search for answers on the internet, it will be counterproductive. Thinking will make you appreciate when you see the solution. Do not look at the comment section before having an answer. You might find it!

- Let us suppose that the chance of having a boy or a girl is exactly 1/2. Also suppose that there is a family with two children. You find out that one of them is a boy. What is the probability that the other child is a girl ? If the problem is ambiguous, then think of randomly selecting a family among all family with two children in which at least one is a boy. What is the probability that the other child is a girl?
- A fair coin i.e., which has probability exactly half of ending with heads (H) and half of ending with tails (T), is tossed five times. Which one of these sequences is more probable a) TTTTT b) HTHHT ?
- Suppose that an HIV test is such that a) if you have HIV then it will be positive with probability 99.9% and b) if you are not infected then the test is negative with probability 99.5%. Suppose your test positive, what is the probability that you are infected?
- Suppose you have two fair dice. What is the chance of rolling a 7 (the sum of outcomes on both dice) given that at least one of them is 4?

:Here Buffett is talking about expected amount of return.Take the probability of loss times the amount of possible loss from the probability of gain times the amount of possible gain. That is what we are trying to do. It is imperfect but that is what it is all about.-- Warren Buffett, 1989 Berkshire Annual Meeting

For example, let us suppose that we throw a fair dice. What is the expected value of the outcome? As each of the face is equally probable, the expected value can be calculated as (1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6), which is equal to 3.5.

The law of large numbers is arguably the most important law to cross over to finance. In its simplest form the law says the following.

:If we toss a fair coin four times then the expected number of heads is 2. But with probability 1/16 you may get four heads. And with probability 1/16 you may get no heads. But if you keep tossing then the number of heads will get closer and closer to the expectation. So, with 1,000 tosses, the number of heads will be closer to 500. With infinite tosses, exactly half of them will be head!If you repeat the same experiment a large number of times then the average of the results will be close to the expected value.

On the risk of oversimplifying, a major consequence is the following. If you pick many stocks, then your portfolio will perform close to the market.

### About the author:

*I invest because I want to be free by the time I reach 40 years of age i.e., 2025. My investment style is to find a small number of bets with large margins of safety. I pay a lot of attention to management and their incentive. Ideally, I like to buy owner operator businesses. I am fortunate to have a strong inclination towards studying. I aid my financial understanding by extensive reading in psychology, economic, social sciences etc.*

**Chandan Dubey**

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