.
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.
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.
So, this artificial intelligence should fix his brain accordingly and consistently.
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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.
It has no feeling of shame - it is shameless.
This time, again, it made an error.
Although the @CPhill' solution are copy-paste Google AI solutions, there is one essential difference.
Every time, Google AI makes a note at the end of its solutions that Google AI is experimental
and can make errors/mistakes.
All @CPhill' solutions are copy-paste of Google AI 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, @CPhill embarrassed to tell the truth.
But I am not embarrassing to tell the truth, as it is my duty at this forum.
And the last my comment.
When you obtain such posts from @CPhill, remember, that NOBODY is responsible for their correctness,
until the specialists and experts will check and confirm their correctness.
Without it, their reliability is ZERO and their creadability is ZERO, 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.