When the PowerBall draw on Tuesday produced a sequence of consecutive numbers it raised an interesting question and accusation from many South Africans — what are the chances?

Statisticians and Ithuba officials pointed out (correctly) that the chance of any sequence of numbers coming up in the PowerBall is exactly the same, **around 1 in 42 million** or 0.000002%.

While this sequence of numbers happens to form a pattern that humans recognise, to the machine that draws these numbers it’s just another random series in the 42,375,200 possible combinations that it can generate on game night.

However, asking what the chances are of seeing Tuesday’s specific sequence — 5, 6, 7, 8, 9, and the PowerBall 10 — is one of the least interesting questions to answer.

The far more interesting question people are asking is: what are the chances of seeing any kind of consecutive sequence in a PowerBall draw?

### Understanding PowerBall

To answer this question we first have to understand the structure of the PowerBall game.

Players must choose five numbers between 1 and 50, and then a sixth bonus or “PowerBall” number between 1 and 20.

This limits the number of consecutive sequences available, as the highest number we can have as the final number in the sequence is 20.

### Basic probabilities

When you are flipping a coin or throwing dice, calculating the probability of one outcome or another is intuitive.

For a completely fair, evenly-weighted coin the chances of either outcome are equal. You have a 1/2 or 50% chance of getting heads, and a 1/2 or 50% chance of getting tails.

It’s similar for a standard six-sided die. There are just more possible outcomes, all of which have equal probability. You have a 1/6 or 16.67% chance of rolling 1, a 16.67% chance of rolling 2, and so on.

When you’re dealing with multiple coin flips or dice throws, a general guideline in statistics is that wherever you find the word “AND”, you multiply. Wherever “OR” appears, you add.

For the purposes of our Lotto calculation, we’ll be doing a lot of multiplication.

Now let’s flip a coin twice and ask some questions:

- What are the chances of getting heads twice in a row? 25%
- What are the chances of getting tails twice in a row? 25%
- What are the chances of getting heads and then tails, or tails and then heads? 50%

### Independent events

One question that catches out lots of people is: What are the chances of getting tails, given that you’ve just flipped the coin and got tails?

The answer is 50% because every coin flip is an independent event. Previous coin flips don’t impact on the probabilities of future coin flips.

This is the trickiness of randomness and statistics — how questions are worded is incredibly important.

One way to think about it is whether you are looking at a combined outcome or an individual outcome. The following two questions are very different:

- Flip a coin twice. What are the chances of both flips coming up tails? (25%)
- You’ve flipped a coin and got tails. What are the chances of getting tails on the next flip? (50%)

Another way to think about it is in terms of an experiment.

Imagine you flip a coin a thousand times. If the coin is perfectly fair, you would expect to see heads come up roughly 500 times and tails 500 times.

However, it is entirely possible for there to have been ten flips in a row which all came up heads, which were cancelled out later in the experiment by ten tail flips in a row.

It is the misunderstanding of this concept of independent events that leads to logical errors like the “**gambler’s fallacy**” and the “**hot hand fallacy**“, as well as the confusion about the likelihood of Tuesday night’s lottery draw.

If every ball is equally likely to come up, then getting a PowerBall draw of 5, 6, 7, 8, 9, and 10 is just as likely as the result 50, 40, 30, 20, 10, and 5.

### Probability in cards and lotteries

When you’re dealing with a deck of cards or pool of lottery balls probability calculations get more complicated, because you remove options as they are dealt from the deck or drawn from the pool.

After getting tails, the tails side of the coin is not erased. If you flip again, you can just as easily get tails as heads.

However, in a deck of cards or pool of numbered lottery balls, if you deal a card or draw a ball, you can’t get that card or ball again until it is shuffled back in.

On a completely shuffled deck, your chances of drawing the Ace of Spades (or any other card) is 1/52 or roughly 1.92%.

When you draw another card it is now impossible to draw the Ace of Spades as the card has been removed from the deck. The number of cards you can draw from has also decreased.

Therefore, on your second draw, your chances of drawing the Queen of Spades (or any other card) is 1/51, or 1.96%.

### Calculating combinations

Calculating the probability of a specific number coming up in the Lotto is similar to the example with the deck of cards.

A simpler way to perform these calculations is with combinations. The concept of combinations is so important in statistics, and mathematics in general, that it has its own notation:

It may look strange and complicated, but the concept is straightforward.

Let’s say you only draw one of the 50 possible PowerBall numbers. Since you can draw any one of the fifty balls, the total number of combinations is 50.

Now let’s draw three balls from the pool. You might be tempted to say that the total number of combinations is 50 × 49 × 48. However, you have to take into account that you can draw the balls in any order.

For example, let’s say the result of drawing three Lotto balls is 7, 8, 9.

There are six different ways you could have got that result:

- First draw the 7, then the 8, then the 9
- 7, then 9, then 8
- 8, 7, 9
- 8, 9, 7
- 9, 7, 8
- 9, 8, 7

You can also calculate this by multiplication rather than manually writing out every possible permutation. The number of ways you can draw three balls from a pool in any order is 3 × 2 × 1 = 6.

To correct for the fact that the order in which you draw the balls doesn’t matter, you must just divide your combination calculation by six: 50 × 49 × 48 ÷ 6 = 19600.

There are therefore 19600 ways of choosing any three balls from a pool of fifty. Stated differently: there are 19600 combinations of three balls out of a pool of fifty, or the odds of getting any sequence of three balls from a pool of fifty is 1 in 19600.

Mathematicians have a short-hand way of expressing that you need to multiply consecutive numbers together, called factorial. It uses the exclamation mark as its symbol.

- 1! = 1
- 2! = 1 × 2
- 3! = 1 × 2 × 3
- …and so forth

The definition of calculating combinations can therefore be expanded as follows:

### Calculating the PowerBall odds

Using the principles explained above, we can calculate the odds of getting any particular sequence of numbers during a PowerBall draw.

First, we choose five balls from a pool of fifty: _{5}C^{50} = 2118760.

Then we factor in the 20 possible PowerBall bonus balls: _{5}C^{50} × 20 = 42375200.

Your odds of guessing the correct sequence to win the PowerBall jackpot is therefore 1 in 42 million or 0.000002%.

### What are the chances of the PowerBall draw being a consecutive sequence?

Finally, we can answer the more interesting question. What are the chances of seeing a sequence of consecutive numbers come up in a PowerBall draw?

As mentioned earlier, the highest number available for the bonus ball is 20. This means there are 15 possible sequences of consecutive numbers:

- 1 2 3 4 5 6
- 2 3 4 5 6 7
- 3 4 5 6 7 8
- 4 5 6 7 8 9
- 5 6 7 8 9 10
- 6 7 8 9 10 11
- 7 8 9 10 11 12
- 8 9 10 11 12 13
- 9 10 11 12 13 14
- 10 11 12 13 14 15
- 11 12 13 14 15 16
- 12 13 14 15 16 17
- 13 14 15 16 17 18
- 14 15 16 17 18 19
- 15 16 17 18 19 20

It is interesting to note that, since the first five numbers can be drawn in any order, there are 1800 (15×5!) ways of drawing any of the above sequences.

However, since our calculation of the PowerBall odds already takes into account that the first five balls can be drawn in any order, we don’t need to compensate for that again.

To calculate the odds of drawing a consecutive sequence of numbers in PowerBall, we simply divide the total possible combinations by the total number of possible consecutive sequences: 42375200 ÷ 15 = 2825013

QED.

*Thanks to Gary for his help in tackling this question.*

### Postscript: Problems at the lottery

While calculating the probability of seeing a specific outcome during a PowerBall draw is an interesting academic exercise, there have been far more pressing issues at the Lotto in the past several years.

Lotto operator Ithuba and Hosken Consolidated Investments (HCI) were **engaged in a protracted legal battle** over the repayment of a R341-million loan, outstanding management fees, and HCI’s rights to have oversight of the lottery operations.

According to reports, the loan had an interest rate of 25% and strict repayment conditions.

Business Day **reported** earlier this year that the matter was resolved after Ithuba agreed to pay HCI a R400-million settlement.

Ithuba agreed to the settlement after HCI was awarded the right to examine Ithuba’s financial statements.

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