Value Number Odds

Generating random numbers has various applications such as random sampling, Monte Carlo methods, board games, and gambling. However, natural processes in physics are typically deterministic, meaning they produce the same outcome every time from the same starting point. Notable exceptions include radioactive decay and quantum measurement, which are truly random processes but not practical sources of randomness. Thus, pseudorandom number generators were developed to produce sequences of numbers that are statistically unpredictable even though they are generated by deterministic algorithms.

For generating pseudorandom numbers, a carefully chosen and confidential random seed is essential to avoid generating the same sequence repeatedly. Nonetheless, in some instances where true unpredictability is critical, physical sources of randomness, like atmospheric electromagnetic noise or radioactive decay, may be used to generate random numbers.

In the past, before modern computing, researchers used various methods to generate random numbers such as dice, cards, and tables of pre-existing random numbers. In the late 1920s, the Cambridge University Press published a table of 41,600 random digits, and in 1947, the RAND Corporation generated random numbers by electronically simulating a roulette wheel.

In computational complexity theory, the notion of pseudorandomness refers to a distribution that is indistinguishable from the uniform distribution with significant advantage. Pseudorandom number generators are often employed to design distributions with desired properties that are pseudorandom against a given model of computation with limited resources. This concept has applications in cryptography.

Permutation Problem 1

Choose 3 horses from group of 4 horses

In a race of 15 horses you beleive that you know the best 4 horses and that 3 of them will finish in the top spots: win, place and show (1st, 2nd and 3rd). So out of that set of 4 horses you want to pick the subset of 3 winners and the order in which they finish. How many different permutations are there for the top 3 from the 4 best horses?

For this problem we are looking for an ordered subset of 3 horses (r) from the set of 4 best horses (n). We are ignoring the other 11 horses in this race of 15 because they do not apply to our problem. We must calculate P(4,3) in order to find the total number of possible outcomes for the top 3 winners.

P(4,3) = 4! / (4 - 3)! = 24 Possible Race Results

If our 4 top horses have the numbers 1, 2, 3 and 4 our 24 potential permutations for the winning 3 are {1,2,3}, {1,3,2}, {1,2,4}, {1,4,2}, {1,3,4}, {1,4,3}, {2,1,3}, {2,3,1}, {2,1,4}, {2,4,1}, {2,3,4}, {2,4,3}, {3,1,2}, {3,2,1}, {3,1,4}, {3,4,1}, {3,2,4}, {3,4,2}, {4,1,2}, {4,2,1}, {4,1,3}, {4,3,1}, {4,2,3}, {4,3,2}

Permutation Problem 2

Choose 3 contestants from group of 12 contestants

At a high school track meet the 400 meter race has 12 contestants. The top 3 will receive points for their team. How many different permutations are there for the top 3 from the 12 contestants?

For this problem we are looking for an ordered subset 3 contestants (r) from the 12 contestants (n). We must calculate P(12,3) in order to find the total number of possible outcomes for the top 3.

P(12,3) = 12! / (12-3)! = 1,320 Possible Outcomes

Permutation Problem 3

Choose 5 players from a set of 10 players

An NFL team has the 6th pick in the draft, meaning there are 5 other teams drafting before them. If the team believes that there are only 10 players that have a chance of being chosen in the top 5, how many different orders could the top 5 be chosen?

For this problem we are finding an ordered subset of 5 players (r) from the set of 10 players (n).

P(10,5)=10!/(10-5)!= 30,240 Possible Orders