Can not get intuition for combined probability, minimum of three

flip101

New member
Joined
Oct 9, 2018
Messages
5
Hello, I am studying a question with it's answer from the book found here https://github.com/theGreenJedi/The-Science-of-Uncertainty/tree/master/Book

Question
Problem 26. PMF of the minimum of several random variables.
On a given day. your gold score takes values from the range 101 to 110. with probability 0.1, independent of other days. Determined to improve your score, you decide to play on three tdifferen days and declare as your score the minimum X of the scores X1, X2 nd X3 on the different days.

(a) Calculate the PMF of X.
(b) By how much has your expected score improved as a result of playing on three days?

Answer
Solution to Problem 2.26.
(a) The possible values of the random variable X are the ten numbers 101, ..., 110, and the PMF is given by

Code:
px(k) =
  P(X > k - 1) - P(X > k), if k = 101, ...110,
  0,                       otherwise.

We have P(X > 100) = 1 and for k = 101, ... 110,
Code:
P(X > k) = P(X1 > k, X2 > k, X3 > k)
         = P(X1 > k)P(X2 > k)P(X3 > k)
         = (110 - k)^3 / 10^3

It follows that

Code:
px(k) =
  ((111 - k)^3 - (110 - k)^3) / 10^3, if k = 101, ...110,
  0,                                  otherwise.

My question
I understand the given answer numerically. But still it feels like magic to me.

I split the possible values of X1, X2, X3 into three sections (on a number line):

Code:
First: X <= k - 1
Second: X > k - 1  and  X <= k
Third: X > k

The first section doesn't count towards the probability that the questions asks. Because it asks X >= k for all three X.
The second section is the probability that the answer is finding.
The third section shows up in the equation with P(X > k).

So as i read it when i have the formula

P(X > k - 1) - P(X > k)

Then P(X > k - 1) is second and third section
and
P(X > k) is the third section

By subtraction P(X > k - 1) - P(X > k) is then just the second section.

I made a venn diagram (with squares) in which the rows are X1, X2, X3 (independent of course). And the columns are the sections on the number line.

1oxeMEb.jpg

If i now want to answer a DIFFERENT question. What is the chance that each X1, X2, X3 is at least K? I put the following formula:

((B or C) - C) * ((E or F) - F) * ((G or I) - I) = B * E * H

This i understand.

When i look at the ORIGINAL question, what is the chance that ONE of X1, X2, X3 is at least K? I put this formula:

((B or C)*(E or F)*(G or I)) - (C * F * I)

Geometrically the two formula's give me the same pieces on the venn diagram.
First formula: first row, second column and second row, second column and third row, second column = second column
Second formula: second column and third column - third column = second column

Like i said before, i understand the original answer numerically. But i'm trying to understand it in different ways (in my case the venn diagram), because i don't find it intuitive at all. Does someone have another way to understand this answer. With venn diagram or another solution alltogether?

thanks for the help
 
Top