You are watching someone throwing a fair coin 4 times and all 4 times coin happens to come head what would be your bet for the fifth time? Are you going to bet head or tail? In other words is the likelihood of a tail increases after 4 heads in a row? Think about it before you read rest of this.
What about the fact that coin does not have any memory so everytime is same as previous one with same probability of 1/2 for head and tail. The answer may lie in binomial distribution. n! ——- [ P^k * (q)^(n-k)] ; q = 1-p = 1/2 in case of fair coin k! (n-k)! The order in this equation does not matter. So in case of fair coin prob of 5 heads in a row = (1/2)^5 while prob of 4 heads is 5 throw regardless of the order is = 5 * (1/2) ^ 5 So odds of getting one tail in 5 throws is 5 times higher than 5 heads in a row.
Moral of the story is that it’s prudent to bet that one out of 5 throws is tail but it’s not prudent to bet that 5th one specifically is tail. We know that if the trials are independent the 5th throw could equally be head or tail.
So basically you can only do better in combinatorics not in the probability sense! 5 heads in a row has the same probability as 4 heads and one tail. But the combinatorics favors the 4 heads and one tail over 5 heads but the tail can come at anytime not necessarily the 5th time!
There is a caveat here though. We are assuming that the coin is fair. But if we don’t know that for sure and see 4 heads in a row you may wonder if the coin is biased toward heads! Hum..
So before applying binomial formula make sure you can safely assume the coin is fair.
To sum up, if you are going to bet a good bet is to say I am willing to say in a 10 throws one or two are going to be tails. But you can never achieve odds in your favor if you are betting on a specific throw!
So let’s get real now. You are in Bellagio checking the hot girl on the black jack table that all of a sudden you notice that the are 9 reds in a row on the roulette table across the room! What are you going to do?
Obviously you will see people gathering around that table and are betting their life on Black! What is an educated mathematician like you to do?
You are not fool anymore and you know that table has no memory and odds of next one being red is as good as the last one. On the other hand there has been 9 reds in a row already!! What the hell?
Think about it. Give me your comments!
