Ask The Wizard #398

Thank you for reminding me. My basic strategy for filling out a bracket is to always pick the higher seeded team (in other words, the lower seed number). When it gets down to a 1 seed vs. 1 seed, pick randomly. That said, adding a 39th year to my March Madness data, here are the probabilities along the way to a perfect bracket.

• 1 seed beats 16 seed = 98.72%
• 2 seed beats 15 seed = 92.31%
• 3 seed beats 14 seed = 85.26%
• 4 seed beats 13 seed = 78.85%
• 5 seed beats 12 seed = 64.74%
• 6 seed beats 11 seed = 60.9%
• 7 seed beats 10 seed = 61.54%
• 8 seed beats 9 seed = 50%
• 1 seed beats 8 seed = 78.75%
• 4 seed beats 5 seed = 55.42%
• 3 seed beats 6 seed = 60%
• 2 seed beats 7 seed = 70.79%
• 1 seed beats 4 seed = 71.01%
• 2 seed beats 3 seed = 60.66%
• 1 seed beats 2 seed = 55.07%
• 1 seed beats 1 seed = 50%

The player must successfully win each these matchups, except 1 vs. 1, four times. In rounds 5 and 6 there would be three 1 seed vs. 1 seed games, which the player must also get right.

The bottom line is the probability of picking all 63 games correctly under this strategy is 1 in 70,166,868,878.

To answer your other question, here is the expected wins per team by seed. For example, any given #5 seed can expect to win 1.153846 games.

1. 3.301282
2. 2.320513
3. 1.839744
4. 1.557692
5. 1.153846
6. 1.057692
7. 0.897436
8. 0.730769
9. 0.596154
10. 0.602564
11. 0.653846
12. 0.50641
13. 0.25
14. 0.160256
15. 0.108974
16. 0.012821

The Prime Number Theorem says a couple of very interesting things:

1. The average distance between primes around the number n is about ln(n).
2. An estimate of the number of primes less than n is n/ln(n).

To to test out the second part, I wrote a program to count the number of primes under one million, two million, up to ten million. The following table shows the number of primes as well as the estimate using the formula above. The right column is the ratio of the estimate to the actual number of primes.

As you can see, the ratio of the number of primes under ten million is 93.4% the actual number. However, this ratio does get smaller as the number of range you're counting for them increases.

For more information, please see the Wikipedia page on the Prime Number Theorem.

To start, let's start with some assumptions about the rules. I'll go with what seems to be the most common ruleset in the U.S.

• Six decks
• Dealer hits soft 17
• Double after split allowed
• Surrender not allowed
• Player may re-split to up to four hands, including aces.

That said, the following list shows the top 20 closest calls according to the player's first two cards and dealer's up card.

This question was asked and discussed in my forum at Wizard of Vegas.