Ask The Wizard #382

Here is my solution (PDF).

This topic has been brought up and is under discussion in my forum located at Wizard of Vegas .

The table below provides the findings from a simulation involving more than 60.5 billion rounds played with a six-deck shoe. The inverse column reflects the inverse probability. For instance, the odds of obtaining a six-card total of 21 are 1 in 280.

This topic has been brought up and is under discussion in my forum located at Wizard of Vegas .

To start, you calculate the expression inside the parentheses, which is 2 + 2 equating to 4. After this step, we arrive at:

8 / 2 * 4 =

Then, we proceed with multiplication and division. If there are multiple operations, we handle them from left to right, kicking off with division. Thus, we compute 8 divided by 2, which gives us 4. At this stage, we have:

4 * 4 = 16

I've encountered a similar question on Facebook, where the most common answer, unfortunately, was incorrectly given as 1. I attribute this mistake to the PEMDAS (Please Excuse My Dear Aunt Sally) guideline, which misleads about the sequence of operations as: parentheses, exponents, division, multiplication, addition, and subtraction.

In truth, division and multiplication are considered on the same level, just as addition and subtraction are. When you face two operations of equal precedence in a single equation, you should work from left to right.

Let’s analyze the amount received by the first five mathematicians:

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.

Observing the trend, we see that f(x) equals f(x-1) divided by (x-1)/100, multiplied by (1 - (x-1)/100), then by (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.

Initially, it’s clear that the portions increase, but at a certain point, they start to decline. The objective is to identify the last mathematician who receives a larger piece than the one before them.

By solving the equation for x, we demonstrate that the first ten mathematicians each receive greater shares than their predecessors. However, since 11 is greater than 10.512, the 11th mathematician receives less than the 10th.

Consequently, mathematician 10 winds up with the largest portion.

Here are the portions allocated to the first 20 (computed 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 particular question has been discussed thoroughly in my forum at Wizard of Vegas .

This problem was taken from Mind Your Decisions .