Probability of sequence of events?

Question

I'm not totally sure how to word this question, so i'll use an example.

If you are trying to choose the letters ABCD in a certain order, (24 permutations), and you have 5 separate goes at getting this right (where the correct permutation changes each time), why, when working out the overall probability of choosing the correct permutation over the 5 goes, would you work out the probability of not choosing the correct permutation and minus it from one, rather than just working it out directly. Hope you understand what i mean.

Answer 1

I think you are asking: what is the probability of choosing the correct sequence from a random set of data, in a certain amount of tries?

Let's assume you pick the sequence, rather than let it be random (instead of flipping a coin, you show it as either heads or tails).  In this case, if there was only one correct sequence and you had 5 tries to get it right, then you would eliminate the incorrect sequences as you discovered them, thus increasing the probability that you will get it right with every turn.

However, since the correct sequence changes as you discover incorrect sequences, then you do not eliminate incorrect sequences as you discover them, because they could be the correct sequence in the next turn.

Therefore, the probability of picking the sequence in each turn is equivelent, however since you only have 5 turns to pick it, your chances of getting it correct diminishes as you proceed.

For example, let's use dice as an example.  You have 1 die, and a computer picks a random number from 1 - 6.  You have 3 rolls.  Each time you roll and do not pick the correct number, the computer picks a new number from 1 - 6, again randomly.

The probability of rolling the number the computer picked is 1/6, every time you roll the die.  If you got to eliminate the number every time it was wrong the probability would be 1/6 for the first roll, 1/5 for the second, and 1/4 in the third.

If you want to find the probability that you can roll the correct number in 3 turns or less, you would say P(x<=3) and use the binomial distribution to figure out what that pobability is.  The binomial distribution shows you the probability of successes in a sequence of a certain amount of yes/no experiments.  Since you could get a success in the first, second, or third experiments, then you would use the cumulative distribution function.  The cumulative will outline the probability of getting the sequence right on the first, second, third... up to i number, where i is the maximum integer (chance) you have to get a successful trial.