UK National, US Mega MM & Euro MM Lotteries.

Winning the Lottery:

“Scientists have calculated that the chances of something so patently absurd actually existing are millions to one.
But magicians have calculated that million-to-one chances crop up nine times out of ten.” - Terry Pratchett, Mort.

Probability ("Million to one chances"):

The mention of "million to one chances" refers to events that are deemed highly improbable or nearly impossible. However, the twist comes in the assertion that these seemingly impossible events do, in fact, happen. Pratchett's statement introduces a touch of the unexpected, what appears impossible might occur against the odds.

Probability ("9 times out of ten"):

The latter part of the quote introduces a statistical perspective, stating that these million to one chances happen "9 times out of ten." This injects a sense of regularity and predictability into the seemingly chaotic world of improbable events. While the future is always uncertain, the quote implies a certain degree of probability, albeit with a humorous exaggeration.

UK National Lottery: link

Calculating the Probability of Winning the Jackpot:

• Players may pick six numbers from a pool of numbers - six different numbers from 1 to 59.
• If you're playing a lottery where you choose 6 numbers from a pool of 59, denoted as (59C6).
• Once the first number has been drawn, 1-59.
• As the first ball is not replaced, there are only 58 possible values for the second one.
• 57 possible values for the third ball, 56 for the fourth, 55 for the fifth and 54 for the last ball.
• In total there are: 59 x 58 x 57 x 56 x 55 x 54 = 32,441,381,280 possible combinations.
• We have to take into account the fact that it does not matter what order the numbers are drawn in.
• So to calculate the number of ways, 6 numbers can be arranged in: 6 x 5 x 4 x 3 x 2 x 1 = 720 permutations.
• This means for a 59C6 lottery, the calculation is: C(59,6) = 59x58x57x56x55x54 / 6x5x4x3x2x1.
• Or 32,441,381,280 / 720 = 45,057,474 different combinations of six numbers.
• Which gives us a 1 in 45,057,474 chance of winning the UK National lottery.

US Mega Millions: link

Calculating the Probability of Winning the Jackpot:

• Players may pick six numbers from two separate pools of numbers - five different numbers from 1 to 70 (the white balls) and one number from 1 to 25 (the gold Mega Ball).
• If you're playing a lottery where you choose 5 numbers from a pool of 70, denoted as (70C5).
• Once the first number has been drawn, 1-70.
• As the first ball is not replaced, there are only 69 possible values for the second one.
• 68 possible values for the third ball, 67 for the fourth and 66 for the last ball.
• In total there are: 70 x 69 x 68 x 67 x 66 = 1,452,361,680 possible combinations.
• We have to take into account the fact that it does not matter what order the numbers are drawn in.
• So to calculate the number of ways, 5 numbers can be arranged in: 5 x 4 x 3 x 2 x 1 = 120 permutations.
• This means for a 70C5 lottery (with 25 mega ball choices), the calculation is: C(70,5) x 25 = 70x69x68x67x66 / 5x4x3x2x1 x 25.
• Or 1,452,361,680 / 120 x 25 = 302,575,350.
• Which gives us a 1 in 302,575,350 chance of winning the US Mega Millions lottery.

Euro Millions Lottery: link

Calculating the Probability of Winning the Jackpot:

• Players may pick seven numbers from two separate pools of numbers - five different numbers from 1 to 50 (the main balls) and two numbers from 1 to 12 (the Lucky Stars).
• If you're playing a lottery where you choose 5 numbers from a pool of 50, denoted as (50C5).
• Once the first number has been drawn, 1-50.
• As the first ball is not replaced, there are only 49 possible values for the second one.
• 48 possible values for the third ball, 47 for the fourth and 46 for the last ball.
• In total there are: 50 x 49 x 48 x 47 x 46 = 254,251,200 possible combinations.
• We have to take into account the fact that it does not matter what order the numbers are drawn in.
• So to calculate the number of ways, 5 numbers can be arranged in: 5 x 4 x 3 x 2 x 1 = 120 permutations.
• In addition two numbers from 1 to 12 (the lucky Stars) have to be chosen.
• If you're playing a lottery where you choose 2 numbers from a pool of 12, denoted as (12C2).
• Once the first number has been drawn, 1-12.
• As the first ball is not replaced, there are only 11 possible values for the second one.
• In total there are: 12 x 11 = 132 possible combinations.
• We have to take into account the fact that it does not matter what order the numbers are drawn in.
• So to calculate the number of ways, 2 numbers can be arranged in: 2 x 1 = 2 permutations.
• This means for a 50C5 lottery, the calculation is: C(50,5) = 50 x 49 x 48 x 47 x 46 / 5 x 4 x 3 x 2 x 1
• Or 254,251,200 / 120 = 2,118,760
• This means for a 12C2 lottery, the calculation is: C(12,2) = 12 x 11 / 2 x 1
• Or 132 / 2 = 66
• This means for a 50C5 lottery and a 12C2 lottery combined, the calculation is: C(50,5) x C(12,2) or 2,118,760 x 66 = 139,838,160.
• Which gives us a 1 in 139,838,160 chance of winning the Euro Millions lottery.

Let's put that into perspective:

• The Royal Society for the Prevention of Accidents says 30 to 60 people get struck by lightning in Britain each year, which would place the odds to be less than 1 in 1,000,000.
• The odds that one will be struck by lightning in the U.S. during one's lifetime are 1 in 15,300.
• Around the world, approximately 2,000 people are struck by lightning every year: World Population, 8,078,000,000 approximately (8.078 billion), Odds = 2,000 / 8,078,000,000. So, the odds of any individual being struck by lightning in a given year are approximately 1 in 2.47 million, or a 1 in 2,470,000 chance!
• Each lottery ticket has the same odds of winning no matter how many you buy. Each one has an independent probability not altered by the frequency of play or how many other tickets you bought for the same draw.
• Buying more tickets certainly increases the overall likelihood of claiming a prize of some sort, even if it's extremely likely to end up below what you spent on the tickets.
• It's important for players to understand the odds and probabilities associated with different lotteries and make informed decisions when participating.

Temporal Possibilities:

Lottery predictions based on previous winning combinations involve analysing historical data, creating a temporal context. Each lottery draw represents a point in time, and by examining the sequence of past draws, one can identify patterns or trends. This aligns with the idea of considering temporal possibilities — understanding the evolving nature of lottery outcomes over time.

Probability:

Lottery predictions inherently involve probability. The likelihood of a specific combination occurring is influenced by the number of possible combinations. By analysing historical data, one aims to discern patterns that might suggest a higher probability for certain numbers or combinations to be drawn. This connects with probability, where events are assessed in terms of their likelihood.

Complex Interplay of Events:

Lottery predictions, like other predictive models, illustrate the complex interplay of events. Each draw is influenced by various factors, including the random selection of numbers, the specific rules of the lottery, and potentially external factors.

While predicting lottery results based on historical data can be an engaging exercise, it's crucial to recognise the limitations. Lottery draws are designed to be random, and the past does not guarantee future results. The complex interplay of factors and the inherent uncertainty make lottery predictions a challenging but intriguing application of temporal possibilities and probability in the real world.

Data Visualisations: UK National Lottery.

The above chart displays the frequency of winning numbers by date, for the last 180 days.

The above chart displays the frequency of winning numbers, for the dates: 19/11/1994 to 25/11/2023.

A Markdown document with R code to analyse, visualise and predict possible future lottery numbers (10 lines), from the data for the UK National lottery: link

The R code in the above Markdown document:

• Aims to predict future lottery numbers by analysing the historical frequencies of winning numbers.
• The code loads lottery data, reshapes it for analysis, and calculates the observed frequencies of each winning number.
• Rather than adhering to a specific probability distribution law, the code fits a probability distribution directly based on the observed frequencies.
• This distribution is then used to generate predictions for the next set of winning numbers.
• The code offers a practical way to make predictions using the available historical data.

A Markdown document with new R code to analyse and predict possible future lottery numbers (10 lines), from the data for the UK National lottery: link

A Markdown document with R code to analyse and predict possible future lottery numbers (10 lines), from the data for the US Mega Millions lottery: link

The R code in the above Markdown document:

• The prediction system looks at the historical data to identify the winning numbers with the highest cumulative frequency.
• If there are ties or no historical data, it generates a random combination.
• This process is repeated for the specified number of predictions (n).
• The result is a list of predicted winning combinations.

A Markdown document with R code to analyse and predict possible future lottery numbers (10 lines), from the data for the Euro Millions Lottery: link

The R code in the above Markdown document:

• Leverages the observed frequencies of winning numbers in historical data to generate predictions for the next set of winning combinations.
• The uniqueness of numbers within each set is ensured to avoid duplicates in the predictions.
• The code is designed to be flexible, allowing you to easily modify parameters such as the number of predictions or the ranges of possible numbers.

Further code explanation is contained within the above Markdown documents.

Conclusion:

• UK National Lottery, probability calculation: In the case of (59C6), where players choose 6 numbers from a pool of 59, the probability of winning the jackpot is 1 in 45,057,474.
• US Mega Millions, probability calculation: In the case of (70C5) x 25, where players select 5 numbers from a pool of 70 and one Mega Ball from a pool of 25, the probability of winning the jackpot is 1 in 302,575,350.
• Euro Millions Lottery, probability calculation: In the case of (50C5) lottery and a (12C2) lottery combined, where players select 5 numbers from a pool of 50 and two numbers from a pool of 12, the probability of winning the jackpot is 1 in 139,838,160.
• Perspective: To put this into perspective, the odds of being struck by lightning in Britain or the U.S. are comparatively lower. Each lottery ticket purchased has an independent probability, emphasising the importance of understanding the odds associated with different lotteries.
• Comparisons: The complexity of these calculations highlights the intricate interplay of events in the lottery world. Understanding the vast number of possible combinations is essential for evaluating the likelihood of winning.
• Probability and the Quirks of Chance: The quote by Terry Pratchett humorously captures the paradox of probability. Events deemed highly improbable may occur more frequently than expected. This paradox is particularly evident in the world of lotteries, where million-to-one chances can seemingly crop up more often than one might anticipate.
• Analysis and Predictions: The data visualisations and predictions are based on historical winning combinations. However, it's crucial to acknowledge the inherent randomness of lottery draws. The complex interplay of factors makes predictions challenging, emphasising the need for a balanced understanding of probability and chance.

Approaching lottery participation becomes an act of conscious navigation through the boundless sea of probabilities, where the understanding of odds becomes a compass guiding one through the cosmic currents. Informed decisions, akin to celestial alignments, shape the trajectory of one's pursuit, offering a glimpse into the potential transformation of life through the elusive jackpot. Amidst the infinity of outcomes, players are encouraged to embrace the profound paradox of the lottery – a delicate balance between the precision of statistical insights and the boundless whimsy of chance, where the pursuit of the extraordinary resides at the intersection of the finite and the infinite. Players are encouraged to approach lottery participation with a clear understanding of the odds, making informed decisions in the pursuit of a potentially life-changing jackpot.

Be informed and of course, be lucky!

A link to my project on Infinity, on Kaggle.

Patrick Ford 🎲