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Message: <blockquote data-quote="Ovinomancer" data-source="post: 8123208" data-attributes="member: 16814">No, because you're making the incorrect assumption that the fumbles would be evenly spread out.  It's a similar mistake to the first one you made earlier in the thread and, to be perfectly fair, an incredibly easy one to make.  Thinking about probability can be a challenge because it's not immediately intuitive at all.So, what you've done here is used a sample of 400 trial with a 5% chance each for the event of interest.  This does average out to 20 events over the trial.  The problem comes when you use this answer for a different question, which is instead what is the probability at least one event of interest will occur in just 4 trials.  This is not the same question as what is the average expected number of events over a larger sample.  In fact, the probability you will have 20 fumbles exactly in 400 trials is just above 9%.  The 95% margins, ie, where you'd expect 95% of all 400 trail samples to end, is between 13 events and 27 events.  There's a 1% chance you have as few as 9 events or as many as 36 events.  This all averages to 20 events, and statistics would build a model were you ran this over an infinite number of trials and get 20% as you answer, but that's not what the real world outputs.So, why, again, is the odds of rolling a fumble in 4 attacks NOT 20%?  To understand, you need to look at how it could work, and figure the probabilities for those.First case is that you have no fumbles -- pretty easy.  Then just one fumble in four, but you have to consider that fumble could occur in four places, and each is it's own probability.  Then 2 fumbles, which can happen 6 different ways.  Then three fumbles which can happen only four ways, then all fumbles, which can happen only 1 way.  Each of these is it's own probability, and it found by multiplying the die probabilities for each position.  Then you can sum the results.  Here it is in a perhaps easier to follow chart, using 95 for not a fumble and 5 for a fumble:<table style='width: 100%'><tr><td>1st Attack</td><td>2nd Attack</td><td>3rd Attack</td><td>4th Attack</td><td>Combined Probability</td></tr><tr><td>95</td><td>95</td><td>95</td><td>95</td><td>81.45%</td></tr><tr><td>5</td><td>95</td><td>95</td><td>95</td><td>4.2869%</td></tr><tr><td>95</td><td>5</td><td>95</td><td>95</td><td>4.2869%</td></tr><tr><td>95</td><td>95</td><td>5</td><td>95</td><td>4.2869%</td></tr><tr><td>95</td><td>95</td><td>95</td><td>5</td><td>4.2869%</td></tr><tr><td>5</td><td>5</td><td>95</td><td>95</td><td>0.2256%</td></tr><tr><td>5</td><td>95</td><td>5</td><td>95</td><td>0.2256%</td></tr><tr><td>5</td><td>95</td><td>95</td><td>5</td><td>0.2256%</td></tr><tr><td>95</td><td>5</td><td>5</td><td>95</td><td>0.2256%</td></tr><tr><td>95</td><td>5</td><td>95</td><td>5</td><td>0.2256%</td></tr><tr><td>95</td><td>95</td><td>5</td><td>5</td><td>0.2256%</td></tr><tr><td>5</td><td>5</td><td>5</td><td>95</td><td>0.00119%</td></tr><tr><td>5</td><td>5</td><td>95</td><td>5</td><td>0.00119%</td></tr><tr><td>5</td><td>95</td><td>5</td><td>5</td><td>0.00119%</td></tr><tr><td>95</td><td>5</td><td>5</td><td>5</td><td>0.00119%</td></tr><tr><td>5</td><td>5</td><td>5</td><td>5</td><td>0.0000625%</td></tr></table>Now, if we don't count the first line (because no fumbles), then the probability of getting at least one fumble in 4 attacks is the sum of the others, or 18.55%.  That's close to 20%, which makes some sense, but isn't, and even if you run 100 trials of 4 events each it won't change per event.  This is because you're asking a different question from "what's the average expected number of fumbles in 400 rolls" to "what's the expects average chance of a fumble in 4 tries".  The iteration doesn't matter to the second question -- it's always going to be 18.55% no matter how many times you do it, because you're asking a different question than the first one.Probability is extremely sensitive to what question is being asked.  Most people don't have the exposure to recognize this, and so think they're asking one question when the probability is answering a different one.  Statistics is the same way (related, but not the same things).  Both tend to hide assumptions, and those assumptions are critical to the answer received.  Make sure that you understand what question you're asking and what the model assumptions of the tool your using (for stats) are or you will get an answer, but it might not be to the question you thought you were asking.  This is the reason there are so many quotes about statistics being a used to lie in plain sight.  Probability, on the other hand, is pretty clean, once you understand how to ask the questions.</blockquote>

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