I'm trying to figure out what happens to the sample space of an event (let's say it's a coin flip) when the probabilities are biased...

Lets say I flip a coin three times.

In the case of equal probabilities I can picture the sample space like a rectangle with area $100$ and each $3$-flip event has $\frac{1}{8}\cdot 100=12.5$ area in the rectangle. I can calculate the probability of an event by just getting the ratio of the areas.

But when the coin is biased, how does each flip affect the "rectangular" sample space? I'm using the rectangle by just trying to visualize the probabilities of the sample space in terms of area.

Hope my question is clear? In other words: How does biased events change the sample space or the rectangle which I use for counting the probabilities?

Thnx =)

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    $\begingroup$ I don't think there is a canonical graphical representation. But observe that for $k=0,1,2,3$ there are $\binom{3}{k}$ events (regions) with probability (area) $p^k(1-p)^{3-k}$. Would be nice to see a natural arrangement for any natural $n$. $\endgroup$ – Eckhard Jan 14 '13 at 12:36

The sample space doesn't have much to do with the probabilities. There are $8$ possible outcomes when flipping a coin three times, so the sample space consists of $8$ individual points and has no real area.

The probability comes into play from assigning probabilities to these points (or to events, in a more advanced setting). They do not have any impact on the sample space.

Sample spaces and a probability assignment to them are distinct concepts that should not be confused with each other.


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