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Ask The Wizard #382

What is the expected number of spins required to get five reds or five blacks in a row?

anonymous

The answer is 4,592,395/118,098 =~ 38.886306 spins.
 

Here is my solution (PDF).

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

Suppose there was a blackjack promotion that offered a huge prize for getting a total of 21 composed of at least five cards. The more the cards in the 21, the greater the prize. If the player had a "21 or bust" strategy, what would be the odds of achieving a 21 by number of cards?

tooncestdc

The following table shows the results of a simulation of over 60.5 billion hands played with a a six-deck shoe. The inverse column shows the inverse probability. For example, the probability of a six-card 21 is 1 in 280.

21 or Bust

Cards in 21 Count Probability Inverse
Bust 52,104,124,978 0.85994880549 1.16
3 4,759,037,984 0.07854520216 13
4 2,557,594,660 0.04221163821 24
5 908,819,311 0.01499954334 67
6 216,326,234 0.00357034086 280
7 38,049,196 0.00062798024 1,592
8 5,220,188 0.00008615622 11,607
9 572,119 0.00000944250 105,904
10 50,292 0.00000083004 1,204,760
11 3,487 0.00000005755 17,375,910
12 192 0.00000000317 315,571,868
13 14 0.00000000023 4,327,842,761
Total 60,589,798,655 1.00000000000

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

8 / 2 * (2+2) =

anonymous

First you evaluate what is in the parenthesis, which is 2+2 = 4. After doing that we're at:

8 / 2 * 4 =

Next, we do multiplication and division. If there are more than one such operation, we go from the left first, which is division. So we evaluate 8/2 = 4 next. After that, we're at:

4 * 4 = 16

I've seen this asked on Facebook where 1 was incorrectly the most frequent answer. I blame the PEMDAS (Please Excuse My Dear Aunt Sally) rule. That incorrectly suggests the order of operations is: parenthesis, exponents, division, multiplication, division, addition, subtraction.

The actual order puts division and multiplication equally as well as addition and subtraction equally. When you see two operations in the same equations of equal priority, go from left to right.

There are 100 mathematicians at a party with one cake. The mathematicians stand in line for a slice.

  • The first mathematician takes 1% of the cake.
  • The second takes 2% of what is left after the first mathematician.
  • The third takes 3% of what is left after first two mathematicians.
  • The fourth takes 4% of what is left after the first three mathematicians.

This continues until the 100th mathematician takes 100% of what is left after the other 99 mathematicians.

Which mathematician will get the most cake? No calculators allowed!

PT

Let's look at how much the first five mathematicians will get:

  1. 1%
  2. 99% * 2%
  3. 99% * 98% * 3%
  4. 99% * 98% * 97% * 4%
  5. 99% * 98% * 97% * 96% * 5%

Let f(x) = cake mathematician x gets.

We can see in the pattern that f(x) = f(x-1) / ((x-1)/100) * (1 - ((x-1)/100)) * (x/100)

Rearranging the terms :

f(x) = f(x-1) * (100/(x-1)) * ((101-x)/100) * (x/100)

Simplifying:

f(x) = f(x-1) * (101-x)/(x-1) * (x/100)

Let y = f(x) where f(x) = f(x-1)

y = y * (101-x)/(x-1) * (x/100)

Divide both sides by y.

1 = (101-x)/(x-1) * (x/100)

100*(x-1) = x * (101-x)

100x - 100 = 101x - x^2

x^2 - x - 100 = 0

Using the Pythagorean formula, x = (1 + sqrt(401))/2 =~ 10.512.

It's obvious the shares increase at first and at some point decrease. The question is to find the last mathematician who gets more than the previous one.

By solving for x above, we show the first 10 mathematicians each get larger shares than the last one. However, since 11 > 10.512, mathematician 11 gets less than mathematician 10.

Thus, mathematician 10 gets the largest share.

Here is the share of the first 20 (using a calculator).

  • Mathematician 1 = 0.01
  • Mathematician 2 = 0.0198
  • Mathematician 3 = 0.029106
  • Mathematician 4 = 0.03764376
  • Mathematician 5 = 0.045172512
  • Mathematician 6 = 0.0514966637
  • Mathematician 7 = 0.0564746745
  • Mathematician 8 = 0.0600245112
  • Mathematician 9 = 0.0621253691
  • Mathematician 10 = 0.062815651
  • Mathematician 11 = 0.0621874944
  • Mathematician 12 = 0.0603784037
  • Mathematician 13 = 0.0575607449
  • Mathematician 14 = 0.0539299902
  • Mathematician 15 = 0.0496926338
  • Mathematician 16 = 0.0450546547
  • Mathematician 17 = 0.0402112793
  • Mathematician 18 = 0.0353386184
  • Mathematician 19 = 0.0305875375
  • Mathematician 20 = 0.0260799004

This problem is asked and discussed in my forum at Wizard of Vegas.

This problem was take from Mind Your Decisions.