What type of math is used in gambling?

Probability Theory Probability Theory As discussed above, probability theory is at the core of mathematics in gambling. Essentially, when one gambles, one is guessing the outcome of an event where it can go a number of probable ways.

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Two ways of working out probability are combinations and permutations.

A permutation is associated with arranging things where the order matters. Combinations are concerned with arranging things where the order does not matter. When it comes to something like horse racing – where the order of finishing the race matters – you can work out the number of arrangements using a permutation formula. When it comes to something like the lottery – where the order in which the numbers are drawn does not matter – you can work out the number or arrangements using a combination formula.

Let’s take a look at this using an example:

Say we have three letters:

A B C

We could arrange them in other ways:

A C B B A C B C A C A B C B A

There are a total of 6 permutations (P) of these three letters.

P = 6

However, the number of combinations here is only one.

C = 1

Application of permutations (without repetition) in horse racing

Let’s say there are four horses that are considered to have an equal chance of winning.

We’ll call them

A B C D

We want to find out all the possible scenarios – or permutations – that these horses could rank (assuming there are no photo-finish ties.)

Let’s start with Horse A coming through as the winner.

There are 6 permutations where Horse A comes first, which means there must also be 6 arrangements where Horse B comes first.

Similarly, there will be 6 permutations for Horses C and D.

Therefore there are 24 arrangements – or permutations – altogether.

Another way of thinking about this is to say we can choose:

The first horse in 4 ways (as there are 4 horses to choose from) The second horse in 3 ways (as there are now only 3 horses to choose from) The third horse in 2 ways (as there are now only 2 horses to choose from) The last horse in 1 way (as there is now only 1 horse to choose from.) So the number of arrangements = 4 × 3 × 2 × 1 = 24 ways Another way of showing this is: 4 × 3 × 2 × 1 = 4! (called 4 factorial – the exclamation mark has this special meaning in math) In general, the number of ways of arranging n distinct (different) objects (where n can be any exact number) is:

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n! (n factorial)

n! = n × (n – 1) × (n – 2) × … × 2 × 1

For example:

The number of ways 7 horses could rank is 5,040. See below – the way we calculate this is just like the example with 4 horses. 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040 A common horse-racing bet is a “perfecta” or “exacta” bet – when you bet on which horses in a race will finish first and second in the exact order.

If there are 5 horses with an equal chance of winning:

We can use the following formula to calculate the number of different permutations possible:

P(n,r) = n! /(n-r)!

where n = the set size (in this instance, the total number of horses) r = the item select size (the number of horses that must finish in a fixed position)

Applying the formula to the horse race will look like this:

P(5,2) = 5!/(5-2)!

P (5,2) = 5!/3! = 20

This means that there are 20 different possible ways for the horses predicted to place in the first and second place in an exact order to fulfill the parameters of the bet. To calculate the probability of winning the bet, the number of arrangements that will win the bet must be divided by the total possible number of arrangements of the horses. To calculate the total number of arrangements possible, we use a factorial as seen before (for a five-horse race we use 5!):

5! = 5 x 4 x 3 x 2 x 1

= 120

Thus the probability of winning an exacta bet in a race with 5 horses that have an equal chance of winning (note that this is simplified, because in real life there are many factors affecting a horse’s chance of winning) is 20 divided by 120.