# Ask The Wizard #398

Now that March Madness is over, did you update your probability of a perfect bracket and average wins by seed?

anonymous

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

What is the Prime Number Theorem?

anonymous

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.

### Primes

Maximum
Number
Total
Primes
Estimate Ratio
1,000,000 78,498 72,382 0.9220925
2,000,000 148,933 137,849 0.9255754
3,000,000 216,816 201,152 0.9277527
4,000,000 283,146 263,127 0.9292967
5,000,000 348,513 324,150 0.9300950
6,000,000 412,849 384,436 0.9311788
7,000,000 476,648 444,122 0.9317618
8,000,000 539,777 503,304 0.9324303
9,000,000 602,489 562,053 0.9328845
10,000,000 664,579 620,421 0.9335545

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.

What are the least costly mistakes in blackjack?

gunbj

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.

### Closest Decisions in Blackjack

Player
Cards
Dealer
Up Card
Best
Play
Second
Best Play
EV 1st
Best Play
EV 2nd
Best Play
Difference
9,7 10 H S -0.535392 -0.536809 0.001417
7,A 2 D S 0.116262 0.113110 0.003152
2,A 5 D H 0.141030 0.137618 0.003412
4,A 4 D H 0.065278 0.060757 0.004521
6,A 2 H D -0.000274 -0.004882 0.004608
10,2 4 S H -0.205906 -0.210664 0.004758
7,2 2 H D 0.073913 0.067870 0.006043
10,6 10 H S -0.534676 -0.540954 0.006278
5,4 2 H D 0.075786 0.068039 0.007747
6,3 2 H D 0.075331 0.067378 0.007953
8,4 3 H S -0.233324 -0.241586 0.008262
7,5 3 H S -0.232183 -0.240505 0.008322
9,2 A D H 0.115609 0.107036 0.008573
3,3 2 P H -0.129464 -0.139266 0.009802
8,A 6 D S 0.462089 0.452220 0.009869
9,3 3 H S -0.237301 -0.248068 0.010767
8,3 A D H 0.118796 0.107445 0.011351
3,3 8 H P -0.219182 -0.230664 0.011482
8,4 4 S H -0.201386 -0.213959 0.012573
9,3 4 S H -0.202651 -0.215698 0.013047

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