Question 1178071
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Roll a die once. Then roll it as many times as the outcome from the first roll. 
Getting the special number "3" on any roll means a win. 
What is the expected number of wins from this experiment? 
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Regarding this problem, I have two notices.


First notice is that the problem's formulation is mathematically incomplete.
To be complete, it should say 


        "Getting the special number "3" on any roll means a win and stopping further rolling".



This correction is almost obvious, but it is important in the analysis.


Second notice is that the solution in the post by @CPhill is INCORRECT.


It is INCORRECT, since there are errors in his analysis.


Below I will show these errors, but I will not provide a whole solution with complete corrections.

Why ?  - - - - Because @CPhill is a pseudonym for the artificial intelligence,
and this solution belongs not to @CPhill, but the artificial intelligence.


So, for AI it will be just a great benefit to get my pointing to their error.
I myself have no any desire to work here for or instead of AI.


So, I will copy-paste here the part of the @CPhill's text with pointing the errors.


<pre>
3.  **Expected Wins Given the First Roll (X):**
    * Let W be the number of wins.

    * If X = 1, W ~ Bernoulli(1/6), E[W|X=1] = 1/6

    * If X = 2, W ~ Binomial(2, 1/6), E[W|X=2] = 2 * (1/6) = 2/6   <<<---=== error. The winning "3" can be obtained at                  
                                                                                    the first of the two rolls (and then the game stops),              
                                                                                    or at the second roll.  It should be correctly counted.

    * If X = 3, W ~ Binomial(3, 1/6), E[W|X=3] = 3 * (1/6) = 3/6   <<<---=== error.  The case of getting X = 3 was just analyzed above,
                                                                                     and it was just led to a stop/break earlier.

    * If X = 4, W ~ Binomial(4, 1/6), E[W|X=4] = 4 * (1/6) = 4/6   <<<---=== similar error: outcome "3" can be obtained in any of 4 rolls,
                                                                                            leading to stop. It should be accounted.

    * If X = 5, W ~ Binomial(5, 1/6), E[W|X=5] = 5 * (1/6) = 5/6   <<<---=== similar error

    * If X = 6, W ~ Binomial(6, 1/6), E[W|X=6] = 6 * (1/6) = 6/6 = 1    <<<---=== similar error.
</pre>

So, this artificial intelligence should fix his brain accordingly and consistently.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . 



Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.


The artificial intelligence is like a baby now. It is in the experimental stage 
of development and can make mistakes and produce nonsense without any embarrassment.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.



This time, again, &nbsp;it made an error.



Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.


Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental
and can make errors/mistakes.


All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:
@PChill never makes this notice and never says that his solutions are copy-past that of Google.
So, he NEVER SAYS TRUTH.


Every time, &nbsp;@CPhill embarrassed to tell the truth.

But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.



And the last my comment.


When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, 
until the specialists and experts will check and confirm their correctness.


Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.




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Hello, @CPhill, don't you think, that it is just a time came to express your thanks to me 
for my tireless work on finding, identifying, pointing and correcting your errors ?


Otherwise, I feel myself uncomfortably about your bad manners.