Do lottery system picks increase/decrease chance of winning?

Question

For a lottery game where:

• The numbers are 1-45
• Standard game is to pick 7 numbers
• 9 numbers drawn (7 winning numbers and 2 supplementary numbers)
• Jackpot = all 7 winning numbers matching, and prizes go down with partial matches (lowest division being 3 winning numbers + 1 supplementary)

If there were three different ways of playing:

1. System 8 (8 games that covers 8 numbers for all 7 number combinations)
2. 8 games that cover all 45 numbers across the games
3. 8 games that are completely random

The simple way of thinking is in each case above 8 games are played, and assuming everything is random and independent, it should mean the probability of winning is the same for all three cases.

However, it also doesn't feel quite right that 8 games that cover a small subset of numbers will have the same probability as 8 games that cover a wide range of numbers.

Is there a difference in the probability of winning (any prize) between the three ways of playing above?

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Concrete example for clarity

Imagine the following three tickets

Ticket A (covers 8 numbers over 8 games)

• 1, 2, 3, 4, 5, 6, 7
• 1, 2, 3, 4, 5, 6, 8
• 1, 2, 3, 4, 5, 7, 8
• etc etc

Ticket B (covers all 45 numbers over 8 games)

• 1, 2, 3, 4, 5, 6, 7
• 8, 9, 10, 11, 12, 13, 14
• 15, 16, 17, 18, 19, 20, 21
• etc etc

Ticket C

• all 8 games are random

If these three tickets were entered for the same draw, would the probability of winning (any prize) be different?

Answer 1

Overlaps increase the probability that multiple "games" will simultaneously win a prize from a partial match. Since the expected number of prizes is the same no matter how you arrange your eight "games," as you increase the probability of multiple prizes you also increase the probability of no prize.

Note that the only case in which "everything is random and independent" is Ticket C; in the other two cases you introduce dependencies in the probabilities of different "games" winning various prizes. With Ticket A you have a higher probability of multiple wins on a partial match, and with Ticket B you (apparently) have a lower probability of multiple wins on a partial match.

(There is some ambiguity about Ticket B, because the pattern you gave can only last for the first six "games"; after that, there are only three unused numbers and you must start duplicating numbers. But I think you can arrange a ticket so that no two "games" have more than one number in common.)