MHB How Does Probability Change with Each Roll of a Loaded Die?

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The discussion centers on calculating the probability function f(x) for a loaded die rolled until an even number appears. The die's probabilities are defined as P(roll is i) = K*i, leading to the conclusion that K = 1/21. The probability f(1) is determined to be 12/21, representing the chance that the first roll is even. Participants are also exploring how to calculate f(2), which involves the probabilities of rolling an odd number first followed by an even number. The conversation emphasizes understanding the relationship between the rolls and the probabilities associated with them.
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What would be f(x) - given the below information?

A die has its six faces loaded so that P(roll is i)=K*x for x=1,2,3,4,5,6. It is rolled until an even number appears. Let X be the number of rolls needed.

K + 2K + 3K + 4K + 5K + 6K = 1

is the correct format so

21K = 1

so

K = \dfrac{1}{21}

f(x) = \dfrac{1}{21}x ??
 
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Jason76 said:
What would be f(x) - given the below information?
A die has its six faces loaded so that P(roll is i)=K*x for x=1,2,3,4,5,6. It is rolled until an even number appears. Let X be the number of rolls needed.

K + 2K + 3K + 4K + 5K + 6K = 1

is the correct format so

21K = 1

so

K = \dfrac{1}{21}

f(x) = \dfrac{1}{21}x ??

Hi Jason76,

Let me take the liberty to rephrase your problem statement a bit, since I believe $i$ and $x$ are supposed to have different meanings.

A die has its six faces loaded so that P(roll is i)=K*i for i=1,2,3,4,5,6. It is rolled until an even number appears. Let X be the number of rolls needed.
Let $f(x)$ be the probability that $x$ rolls are needed until an even number appears.


Does that look right to you?

It would mean that f(1) is the probability that the very first roll is even, after which we stop.

So:
f(1) = P(1st roll is even) = P(roll is 2 or 4 or 6) = P(roll is 2) + P(roll is 4) + P(roll is 6) = 2/21 + 4/21 + 6/21 = 12/21

and:
f(2) = P(1st roll is odd and 2nd roll is even) = P(1st roll is odd) P(2nd roll is even)

What would f(2) be?
 
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