Good Luck!
In Brazil we have a type of lottery called "Mega Sena". To win it you need to choose 6 numbers between 01 and 60.
If the 6 winning numbers are the same as you choose you get much money!
A regular deck of cards have 52 cards. You can shuffle these cards in many ways. It can start with the Queens, Kings, Aces, Clubs, Diamonds etc.
The question is:
What is more probable to happen with you: Win 5 times in a row in Mega Sena or Play with a deck of cards in the same order as you played last week (in this case the cards will be shuffle at random, both in this game as in the last)?
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2 solutions
Moderator note:
We just need to make sure your bounds are in the right direction. Your initial calculations show that we have a lot of leeway, and so we can choose larger bounds which make the inequalities easier to deal with. For example, we can show that ( 6 6 0 ) < 2 5 6 and then 2 5 3 0 < 5 2 ! .
The total ways that 6 numbers can be selected between 01 and 60 is 50,063,860.
Thus the chance to win in Mega Sena is 1 between 50,063,860.
To win 5 times in a row you will play 5 games with the same chance to win.
How these games are in a row you will have 5 0 , 0 6 3 , 8 6 0 5 1 = 3 . 1 4 5 0 0 7 2 9 1 3 9 3 4 4 7 9 7 5 3 5 9 2 0 7 9 2 6 3 7 4 1 7 6 × 1 0 3 8 1 chance to achieve this feat.
The number of ways that a deck of cards can be shuffled is 5 2 ! . If you play with the cards shuffled in one way, the chance to play with this same shuffle next week is 1 between 5 2 ! .
5 2 ! 1 = 8 . 0 6 5 8 1 7 5 1 7 0 9 4 3 8 7 8 5 7 1 6 6 0 6 3 6 8 5 6 4 0 3 7 6 6 9 7 5 2 8 9 5 0 5 4 4 0 8 8 3 2 7 7 8 2 4 × 1 0 6 7 1 .
How we can see, the chance to win 5 times in a row in Mega Sena is higher than the chance to play with a deck of cards shuffled in the same order as you palyed in a last game.
This is the "Statistical Answer" for this question. It can be a bit complex to understand. If you have another answer, not too complex (like an approach), share with us!
This is not rigorous, but it should be fairly easy to make it so.
There are 6 ! 6 0 ⋅ 5 9 ⋅ … ⋅ 5 5 possibilities for Mega Sena, and only one of them wins. Note that 6 ! ≈ 3 6 and 6 0 ⋅ 5 9 ⋅ … ⋅ 5 5 ≈ 6 0 6 , so there are approximately 2 0 6 possibilities for Mega Sena. In other words, winning Mega Sena is approximately equally likely to guessing a 20-sided die correctly six times in a row. Winning five times of Mega Sena in a row means guessing a 20-sided die correctly 30 times in a row.
However, this is still much smaller than guessing the deck (having the deck exactly identical to a predetermined configuration). The first card of the deck has 52 possibilities; the second card has 51 possibilities; the third card has 50 possibilities, and so on until the 30th card which has 23 possibilities. All of them are greater than 20 possibilities. Thus guessing the first 30 cards of a deck is already harder than guessing Mega Sana, much less guessing the entire deck correctly.