Ask The Wizard #398

I appreciate the reminder. My fundamental approach when completing a bracket involves consistently selecting the team with the higher seed (essentially, the one with a lower seed number). When faced with a match-up between two 1 seeds, I prefer to make a random choice. Incorporating a 39th year into my March Madness statistics, here are the probabilities associated with achieving 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%

A participant must accurately predict the outcome of each matchup, apart from the 1 vs. 1 scenarios, a total of four times. During the fifth and sixth rounds, there will be three games featuring 1 seeds against each other, which also need to be correctly guessed.

Ultimately, the likelihood of correctly picking all 63 games following this strategy stands at 1 in 70,166,868,878.

In response to your other inquiry, here is the anticipated number of wins for each team, arranged by seed. For instance, a #5 seed can typically be expected to secure about 1.153846 victories.

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 offers several intriguing insights:

1. On average, the spacing between prime numbers near the value of n is roughly ln(n).
2. Additionally, an approximation of the quantity of primes less than n is expressed as n/ln(n).

To explore the latter claim, I developed a program to tally the number of primes from one million up to ten million. The table below lists these counts alongside the projections derived from the aforementioned formula. The rightmost column illustrates the ratio of estimated values to the actual prime counts.

As demonstrated, the ratio of primes below ten million is 93.4% of the actual count. Nevertheless, this ratio tends to decrease as you expand the range of numbers being evaluated.

For further details, visit the relevant Wikipedia page on this topic. Prime Number Theorem .

To begin, we’ll outline some basic assumptions regarding the rules. I will base these on what appears to be the most prevalent ruleset utilized in the U.S.

• Six decks
• Dealer hits soft 17
• Double after split allowed
• Surrender not allowed
• Players have the option to re-split their hands up to four times, which includes aces.

With that in mind, the following list presents the top 20 nearest calls based on the first two cards dealt to the player and the dealer's visible card.

This topic was raised and deliberated on my forum at Wizard of Vegas .