Probability and Odds

Probability is human’s way to understand uncertain events.

Some of knowledge of Probability is required to play the game of probability (the stock market).

Flip a coin, the probability to get a Head is 50%.

If the coin is biased, say, probability to get a Head is 60%, probability to get a Tail is 40%. Now the question is that how do you design a fair bet?

One answer is to bet the sequence of Head and Tail, one bets “HT” (head, then tail), and the other bets “TH”, and you flip the coin twice.

The theory behind it is that the probability to get “HT” and “TH” is the same.

P(HT)=P(H)*P(T)=60%*40%
P(TH)=P(T)*P(H)=40%*60%

Len Deighton in his famous novel “Bomber” talked about a World War II pilot had two percent chance of being shot down on each mission. Now the question was what is the probability of being shot down on 50 missions?

The novelist says the flyer is “mathematically certain to be shot down”. In other words, he thinks the probability is 100%.

That is very wrong, my friend, the correct answer is: 1-(0.98)^50 = 0.64, i.e., 64% chance being shot down on 50 missions.

Now what is the probablity of Head for the next coin-tossing after you saw 10 consecutive tails?, if you think the Head is more possible after 10 Tails, you are fallen into the trap called "Gambler's fallacy".

The "birthday problem", what is the probabilty of at two people share the same birthday among a group of 23 people in a party?. The answer will surprise most people.

P=1-(364/365)*(363/365)*(362/365)....*(343/365)=0.5073

That is saying that among 23 people in any gatherings, there is 50.73% probabilty that at least ywo people share the same birthday.

Now let's say you have two trading strategies, both winning percent is 60%. You take 2 trades, one from each trading strategy.
(1) What is probabilty of both trades are winners?
(2) What is probablity of at least one winner?
(3) What is probabilty you have winner(s)?

Answer (1) P=60%*60%=36%
(2) P=60%*(1-60%) + 60%*(1-60%)=48%
(3) P=36% + 48% =84%



“Odds” is not probability. “Odds” is often used by gambling industry. Odds is defined as
PayOff –to-cost, when people say the odds is 100-to-1, that means, you bet $1 to win $100. Now what is the probability to win this kind of game?

Assume it is the fair game (Edge is zero), then the probability for a given Odds is
P= Cost/ (PayOff + Cost)

Thus, probability to win a Odds 100-to-1 game is P=1/(100+1)=1/101, which is less than 1%.

Odds is a useful term when we talk about position sizing using Kelly’s formula.