Why learn probability and counting for 4D and TOTO? The rules of each game determine what counts as a possible result and a match. Probability helps us describe how likely a result is; combinatorics helps us count the possibilities. Learning these basics makes it easier to understand the odds and historical statistics in the game-specific guides.

1. Probability

Probability is a measure of how likely an event is to occur. We express it as a fraction, a decimal, or a percentage. For example, if we flip a fair coin, there are two possible outcomes: heads or tails. Since it is fair, the probability of getting heads is 1/2, and the probability of getting tails is also 1/2.

In general, the probability of an event (e.g. heads) is the number of ways it can happen divided by the total number of possible outcomes (e.g. heads or tails). This formula works when each outcome is equally likely, which is often assumed in games of chance like lotteries and dice rolls.

Let's see how this works with one more simple example. A fair six-sided dice has six possible results. Only one is a 4, so the chance of rolling a 4 is 1/6. The chance of rolling an even number is 3/6, or 1/2, because 2, 4 and 6 all count.

Flip a fair coin

Heads Tails

Chance of heads: 1 of 2 = 1/2

Roll a fair six-sided die

1 2 3 4 5 6

Chance of an even number: 3 of 6 = 3/6 = 1/2

Highlighted outcomes are the ones that fit the event. With a fair coin or die, each outcome is equally likely.

In general, if every outcome is equally likely, the chance of an event is:

P(event)= number of outcomes that fit the 'event'total number of possible outcomes

These fractions describe the chance before a roll. If the dice has not shown a 4 for a long time, that does not make the next roll “due”. Assuming the dice is fair and each roll is independent, what statistics tell us is that given infinite rolls, the proportion of 4s will approach 1/6. But we do not have infinite rolls, so short-term deviations from the expected probability are normal. Certainly, that is hinged on a huge assumption of a 'fair' die, which may not always hold in real-world scenarios (e.g. wear and tear). These are special situations that we might revisit in future articles.

2. Combinatorics: counting without listing

Combinatorics is the part of mathematics that helps us count things. Imagine three different cards: red, blue, and green. We want to choose two cards. How many different results are possible?

For such simple examples, we could list all outcomes. For more complicated cases such as TOTO, combinatorics give us tools to work it out without the infeasible task of writing every outcome down.

Going back to our three cards example, the answer depends on whether the order of the cards matters. If the order matters, we are looking at permutations. If the order does not matter, we are looking at combinations.

Order matters: two arrangements

RedthenBlue

BluethenRed

Order does not matter: one selection

RedandBlue

Naming Blue first still gives the same pair.

The cards are the same; it is the question we ask that determines whether their order counts.

3. Permutations: when order matters

Suppose we place two cards in a first and second position. Red first and blue second is a different arrangement from blue first and red second. These arrangements are called permutations.

There are three choices for the first position. After choosing one card, there are two cards left for the second position. So there are 3 × 2 = 6 arrangements:

Red–blue, red–green, blue–red, blue–green, green–red, green–blue

This uses a useful counting rule: when a task has several steps, multiply the number of choices at each step.

4. Combinations: when order does not matter

Now imagine we are simply taking two cards home. We are not putting them in first and second positions, so red–blue is the same pair as blue–red. When we only care which cards were chosen, not their order, we call each result a combination.

The six arrangements above reduce to just three pairs: red and blue, red and green, or blue and green. Each pair appeared twice in the arrangement list—once in each order—so the number of pairs is 6 ÷ 2 = 3.

5. A shortcut for counting combinations and permutations

Listing every possibility works for three cards, but gets tedious with more. Suppose five friends are available, and we need to choose two to play on a team. If we picked a first and a second person, there would be 5 × 4 = 20 ordered picks. But each team is counted twice: Alex and Sam is the same team as Sam and Alex. So there are 20 ÷ 2 = 10 different teams.

For larger selections, the number of ways to choose k items from n distinct items is written “n choose k” and calculated with this formula:

= n!k!(n − k)!

The exclamation mark means factorial: 4! = 4 × 3 × 2 × 1. For choosing two friends from five:

= 5!2!3! = 10

That matches the ten different teams we counted by removing the duplicate orders.

What about permutation?

For n distinct items arranged in r positions, the permutation formula is:

P(n, r)= n!(n − r)!

For our three cards arranged in two positions, P(3, 2) = 3! / 1! = 6.

6. Linking back to the games

For ordinary 4D, order clearly matters: the number 1516 is different from 1651. In the case of iBet, you would be familiar that order does not matter since a winning number of 1651 could be matched by 1516.

For TOTO, order does not matter. A draw of 3, 8, 15, 21, 34 and 47 is the same six-number set regardless of which ball came out first.

Ready to apply these ideas? Work through the TOTO odds and see how combinations count each prize group, or continue to the separate 4D odds guide.

Go further

Khan Academy: Statistics and Probability has free videos and practice, including probability and counting topics. OpenStax Introduction to Statistics is a free textbook for a more structured course.