Gambling Guide
Ask the Wizard #324
Ace2
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Here is my solution. (PDF)
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There is a square dartboard of dimensions 1 by 1. A dart is thrown at it such that it can land anywhere with equal probability. Let the coordinates of where is lands be (x,y), where both x and y are uniformly and independently distributed from 0 to 1.
Let z = round(x/y). In other words, z = x/y, rounded to the nearest integer. What is the probability that z is even?
"Anonymous" .
It will be very useful to know the infinite series in the following hint.
The Leibniz formula for π states:
1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = π/4
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If x/y < 0.5, then the ratio will round dow n to 0, and even number. Any point on the dartboard on the left of the line formed by (0,0) and (0.5) will round down to 0. That area is a right triangle with side of 1 and 1/2. Remember the area of a triangle is (1/2)*base*height. Thus the area of those points rounding down 0 to is (1/2)*(1/2) = 1/4.
The next area on the graph that will round to the next even number, 2, is when 1.5 < x/y < 2.5. This area will be a triangle with base 2/3 - 2/5 and height 1. Note these are the inverses of the bounds of x/y, because x equals 1, so we need to invert y. So, the area that rounds to 2 is (1/2)*(2/3 - 2/5).
The next area on the graph that will round to the next even number, 4, is when 3.5 < x/y < 4.5. This area will be a triangle with base 2/7 - 2/9 and height 1. So, the area that rounds to 2 is (1/2)*(2/7 - 2/9).
The next area on the graph that will round to the next even number, 6, is when 5.5 < x/y < 6.5. This area will be a triangle with base 2/11 - 2/13 and height 1. So, the area that rounds to 2 is (1/2)*(2/11 - 2/13).
Starting to see a pattern? It goes:
1/4 + 1/2*(2/3 - 2/5 + 2/7 - 2/9 + 2/11 - 2/13 + ... ) =
1/4 + (1/3 - 1/5 + 1/7 - 1/9 + 1/11 - 1/13 + ... ) =
Let's move a -1 inside those parenthesis.
5/4 + (-1 + 1/3 - 1/5 + 1/7 - 1/9 + 1/11 - 1/13 + ... ) =
5/4 - (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 + ... ) =
Next, recall our hint above:
1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11
Getting back to the question at hand ...
5/4 - π/4 =
(5 - π) / 4 = apx. 0.464601836602552.
Interesting how π and e keep popping up all over the place in math.
This question is asked and discussed in my forum at Wizard of Vegas.
Let 9x + 12x = 16x
What is x?
"Anonymous" .
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9x + 12x = 16x =
Divide both sides by 9x
1 + (12/9)x = (16/9)x
1 + (4/3)x = ((4/3)x)2
(1) Let u = (4/3)x
1 + u = u2
By the quadratic formula...
u = (1+sqrt(5)) / 2 (The Golden Ratio)
Putting that back in equation (1):
(4/3)x = (1+sqrt(5)) / 2
Take the log of both sides:
x ln(4/3) = ln[(1+sqrt(5)) / 2]
x = ln[(1+sqrt(5)) / 2] / ln(4/3)
x = [ln(1+sqrt(5) - ln(2)] / [ln(4) - ln(3)] = apx. 1.67272093446233.
This question is raised and discussed in my forum at Wizard of Vegas.
Acknowledgement: I got a variation of this problem from Presh Talwalkar of Mind Your Decisions.
Suppose a fair six-sided die is rolled until a 1, 2, 3, or 6 appears. If a 1, 2, or 3 is the first of these game-ending numbers to appear, then you win nothing. If a 6 is the first of these game-ending numbers to appear, then you win $1 for every roll of the die. What is the average win of this game?
Klopp
Click the button below for a couple infinite series formulas that you may find helpful.
Hint 1: Sum for i = 0 to ∞ of ni = 1 / (1-n)
Hint 2: Sum for i = 0 to ∞ of i × ni = n / (1-n)2
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Suppose a fair six-sided die is rolled until a 1, 2, 3, or 6 appears. If a 1, 2, or 3 is the first of these game-ending numbers to appear, then you win nothing. If a 6 is the first of these game-ending numbers to appear, then you win $1 for every roll of the die. What is the average win of this game?
Hint 1: Sum for i = 0 to ∞ of ni = 1 / (1-n)
Hint 2: Sum for i = 0 to ∞ of i × ni = n / (1-n)2
The expected win can be expressed as the sum for i = 0 to ∞ of (1 + i) * (1/3)i * (1/6). =
(1/6) * sum for i = 0 to ∞ of (1/3)i + (1/6) * sum for i = 0 to ∞ of (i * (1/3)i).
Let's evaluate these one at a time.
sum for i = 0 to ∞ of (1/3)i =
1 / (1 - (1/3)) =
1 / (2/3) =
3/2
Sum for i = 0 to ∞ of (i * (1/3)i) =
(1/3) / (1 - (1/3))2 =
(1/3) / (4/9) =
(1/3) * (9/4) =
3/4
Putting it all together, the answer is
(1/6) * (3/2) + (1/6)*(3/4) =
(1/4) + (1/8) =
3/8
This question is asked and discussed in my forum at Wizard of Vegas.
