I came across Probability when I was in my 9th Grade. The simple definition is
P(Event) = Number of ways it can happen/Total number of outcomes
I understood how to solve Dice Rolling and Coin Tossing problems. Beyond that I never bothered to think deeply about this. In fact my understanding was very superficial just to get through my exams.
Chance has no memory
I used to watch a lot of Cricket during my school days. I used to keep track of which team won the toss in each game and I used this data to predict who will win the toss in the next game. The idea behind this was if one team lost the toss in the previous games then according to probability it should win the toss in the next game. I knew how stupid I was. Later in my life I read about “Chance has no memory” in this excellent book “Seeking Wisdom From Darwin To Munger”
We tend to believe that the probability of an independent event is lowered when it has happened recently or that the probability is increased when it hasn’t happened recently. For example, after a run of bad outcomes in independent events that appear randomly, we sometimes believe a good outcome is due. But previous outcomes neither influence nor have any predictive value to future outcomes. There is neither memory nor a sense of justice.
The expected value of an uncertain event is the sum of the possible payoffs multiplied by each payoff’s chance of occurring.
Let us understand this with an example. A lottery has 100 tickets. Each ticket costs $10. The cash price is $500. What is the expected value
Chance of winning the lottery = 1/100(only 1 winning ticket) Chance of not winning the lottery(losing $10) = 99/100 Expected value = (1/100) * 500 + (99/100) * (-10) = 0.01 * 500 + 0.99 * (-10) = 5 - 9.9 = -4.9
Why should you bother about Expected Value? The simple reason is it will help you decide whether you want to take the bet or not. In the above example I will not because I will be losing $4.9 on an average if I will play the game for a very long time.
Most of our decisions in every-day life are one-time bets. Choices we face only once. Still, this is not the last decision we make. There are large number of uncertain decisions we make over a lifetime. We make bets every day. So if we view life’s decisions as a series of gambles, we should use expected value as a guide whenever appropriate. Over time, we will come out better.
Some of the decisions cannot be made using the results of Expected value. For example in Russian Roulette there are 6 equally likely possible outcomes when you pull the trigger – Empty,Empty,Empty,Empty,Empty,Bullet. If you survive you win $10 million.
Expected Value = (5/6) * 10 million + (1/6) * 0 = 8.33 million
Even though the probability favor’s you, the downside is unbearable(death). Why should you risk your life? Before taking any decision I look at the downside to make sure that it is bearable. If not I will not play the game even though the probability is in my favor. Remember “Long Term Capital Management“.
The chance of not having something is 1 minus the chance of having it.If the chance of getting a 2 in a single die roll is 1/6 and the chance of not getting a 2 is (1 – 1/6) = 5/6
The chance of two independent events both happening(AND) is the product of their individual probabilities. To explain this principle I am using the problem given by Professor Shai Simonson
Review Problem: You are sitting at home watching the NBA finals. Shaq is on the freethrow line with no time left on the clock. His freethrow shooting percentage is a dismal 68%. He needs to make both shots to win the game. Your Uncle says that he bets you $20 they don’t win the game, and he will give you 2:1 odds. Do you take his bet?
Solution: The chance Shaq will make both shots is .68 x .68 = .4624. If you made this $20 bet 10,000 times then you would expect to win 4624 times, and lose 5376 times. You win $40 4624 times, and you lose $20 5376. This is a net gain of $77,440. If you make the bet once, you have an expected gain of about $7.74. Take the bet!
When two events A and B are mutually exclusive(unable to be both true at the same time), the probability that A OR B will occur is the sum of probability of each event. A single 6 sided die is thrown. What is the probability of getting 2 or 3. P(2) = 1/6 ; P(3) = 1/6. Hence Probability of getting 2 or 3 is 1/6 + 1/6 = 1/3
Sounds simple right? Try this problem. I can throw a crumpled piece of paper across the class into a garbage can 1 in 5 times. If I try five times, what’s the chance I get it in at least once?
P(throwing across class) = 1/5 P(Getting it once out of 5 tries) = 1/5+ 1/5 + 1/5 + 1/5 + 1/5 = 1 (100%) (Strange!) Let us extend this logic for 10 tries P(Getting it once out of 10 tries) = 10 * 1/5 = 2(200%)(Absurd!)
Explanation given by Professor Shai Simonson
This leads to a non-principle of probability – the addition non-principle. It is not true that: The chance of at least one of two or more events occuring equals the sum of the chances that each event occurs. Please never use this hideous illogical but tempting non-principle.
Now let’s do it right. This hard problem requires some careful thinking, a good plan, and the use of our two main principles. The chance of missing a throw is 4/5 (Complement Principle). The chance of missing all five throws is 4/5 x 4/5 x 4/5 x 4/5 x 4/5 = .32768 (Multiplication Principle). The chance of not missing all five, or equivalently the chance of making at least one, is 1 – .32768 = .67232 or about 67%.
When solving problems on Probabilities do not substitue numbers in the formula blindly. Understand what those numbers mean and we can avoid a lot of mistakes.
Everything that can be counted does not necessarily count; everything that counts cannot necessarily be counted – Albert Einstein