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When Tuesday’s PowerBall drawing produced a sequence of consecutive numbers, it generated an interesting question and an indictment from many South Africans: What are the chances?
Ithuba statisticians and officials (correctly) pointed out that the probability of a sequence of numbers appearing in the PowerBall is exactly the same, about 1 in 42 million or 0.000002%.
While this sequence of numbers forms a pattern that humans recognize, for the machine that draws these numbers it is just another random series out of the 42,375,200 possible combinations it can generate on game night.
However, asking what are the chances of seeing the specific sequence on Tuesday (5, 6, 7, 8, 9 and PowerBall 10) is one of the least interesting questions to answer.
The much more interesting question people ask themselves is: what are the chances of seeing some kind of consecutive sequence in a PowerBall draw?
Understanding PowerBall
To answer this question, we must first 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, since the highest number we can have since the final number of the sequence is 20.
Basic probabilities
When you flip a coin or roll the dice, calculating the probability of one outcome or the other is intuitive.
For a completely fair and 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 is similar for a standard six-sided die. There are only more possible outcomes, all of which are equally likely. You have a 1/6 or 16.67% chance of rolling 1, 16.67% chance of rolling 2, and so on.
When it comes to multiple coin or dice tosses, a general guideline in statistics is that wherever you find the word “Y”, it multiplies. Wherever “OR” appears, add.
For the purposes of our Lottery calculation, we will be doing a lot of multiplication.
Now, let’s flip a coin twice and ask a few questions:
- What are the chances that it will come up heads twice in a row? 25%
- What are the chances of getting tails twice in a row? 25%
- What are the chances that it will come up heads and then tails or tails and then heads? fifty%
Independent events
A question that catches the attention of many people is: What are the chances of a tails coming up, given that you have just flipped the coin and tails are coming out?
The answer is 50% because each coin flip is a separate event. Past coin flips do not affect the odds of future coin flips.
Here’s the randomness and stats trick – how questions are asked is incredibly important.
One way to think about it is whether you are viewing a combined result or an individual result. The next two questions are very different:
- Flip a coin twice. What are the chances that both spins will come out tails? (25%)
- You flipped a coin and came up tails. What are the chances of tails on the next pitch? (fifty%)
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, expect heads to appear about 500 times and tails to 500 times.
However, it is quite possible that there were ten spins in a row that all came heads, which were canceled later in the experiment with ten tail spins in a row.
It is the misunderstanding of this concept of independent events that leads to logical errors such as “gambler’s fallacy” and the “hot hand fallacy“As well as the confusion over the probability of the Tuesday night lottery draw.
If all balls are equally likely to come out, then getting a PowerBall tie of 5, 6, 7, 8, 9, and 10 is as likely as the outcome of 50, 40, 30, 20, 10, and 5.
Probability in cards and lotteries
When it comes to a deck of cards or a group of lottery balls, the probability calculations get more complicated because you eliminate the options as they are dealt from the deck or drawn from the pool.
After getting tails, the cross side of the coin is not erased. If you flip again, you can get tails as easily as heads.
However, in a deck of cards or a group of numbered lottery balls, if you deal a card or draw a ball, you will not be able to get that card or ball again until it is shuffled again.
In a fully shuffled deck, your chances of drawing the Ace of Spades (or any other card) are 1/52 or about 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) are 1/51, or 1.96%.
Calculating combinations
Calculating the probability that a specific number will appear in the lottery is similar to the example of the deck of cards.
An easier way to do 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 seem strange and complicated, but the concept is simple.
Let’s say you only roll one of the 50 possible PowerBall numbers. Since you can draw any of the fifty balls, the total number of combinations is 50.
Now let’s get three balls out of the pool. You may be tempted to say that the total number of combinations is 50 × 49 × 48. However, you should be aware 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 to get that result:
- First draw 7, then 8, then 9
- 7, then 9, then 8
- 8, 7, 9
- 8, 9, 7
- 9, 7, 8
- 9, 8, 7
You can also calculate this by multiplying instead of manually typing in all the possible permutations. The number of ways you can get three balls out of a pool in any order is 3 × 2 × 1 = 6.
To correct the fact that the order in which you draw the balls doesn’t matter, you need to divide the combination calculation by six: 50 × 49 × 48 ÷ 6 = 19600.
Therefore, there are 19,600 ways to choose three balls from a group of fifty. In other words, there are 19,600 combinations of three balls out of a group of fifty, or the odds of getting any sequence of three balls out of a group of fifty are 1 in 19,600.
Mathematicians have a shortened way of expressing that it is necessary to multiply consecutive numbers, called a factorial. Use the exclamation point as a symbol.
- 1! = 1
- 2! = 1 × 2
- 3! = 1 × 2 × 3
- …Etc
Therefore, the definition of calculation of combinations can be extended as follows:
Calculate PowerBall Odds
Using the principles explained above, we can calculate the odds of getting a particular sequence of numbers during a PowerBall draw.
First, we choose five balls from a group of fifty: 5Cfifty = 2118760.
Then we take into account the 20 possible PowerBall bonus balls: 5Cfifty × 20 = 42375200.
Your odds of guessing the correct sequence to win the PowerBall jackpot are therefore 1 in 42 million or 0.000002%.
What are the chances that the PowerBall draw will be a consecutive sequence?
Finally, we can answer the most interesting question. What are the chances of seeing a sequence of consecutive numbers appear in a PowerBall drawing?
As mentioned above, the highest number available for the bonus ball is 20. This means that there are 15 possible consecutive number sequences:
- 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 to draw 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, you do not need to compensate again.
To calculate the probabilities of drawing a consecutive sequence of numbers in PowerBall, we simply divide the total number of possible combinations by the total number of possible consecutive sequences: 42375200 ÷ 15 = 2825013
QED.
Thanks to Gary for his help in addressing this question.
Postscript: Lottery Trouble
While calculating the probability of seeing a specific result during a PowerBall draw is an interesting academic exercise, there have been much more pressing issues at Lotto in recent years.
Lottery operator Ithuba and Hosken Consolidated Investments (HCI) were engaged in a protracted legal battle on repayment of a R341 million loan, outstanding management fees, and HCI’s rights to oversee lottery operations.
The loan reportedly carried a 25% interest rate and strict repayment terms.
Working day reported Earlier this year, the matter was resolved after Ithuba agreed to pay HCI a settlement of 400 million rand.
Ithuba accepted the agreement after HCI obtained the right to examine Ithuba’s financial statements.
Now Read: Why South Africans Are Angry About Graphics Card Prices
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