David W. answered • 07/04/15
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It's the form of the question that confuses most people.
Let's break it apart:
"If I roll the dice 10 times and my number doesn't show up"
means O.K. that's history, now get on with it.
"what happens to my odds on the 11th roll"
means O.K. you have to start somewhere, so let's start.
IMPORTANT TO STATISTICS: For "fair dice," the outcome on any one roll does not depend on other rolls.
However, if you want to know the result at the end of multiple rolls, you have some math to do.
"moving forward until it does hit" sounds like wanting to know how many rolls it will take to guarantee that your desired outcome occurs. Well, I hate to tell any gambler, it is theoretically possible (though very highly unlikely) that you could continue to roll the dice and never arrive at "until it does hit." Sorry, that's probability. We don't see such things using experiments (experimental probability) often, so it may be tough to accept.
In the long run, statistics expects experimental probability to resemble theoretical probability (e.g., for 100,000 rolls of one die, it is possible to get 100,000 "2's" with a "fair die," but it is extremely unlikely, we expect that the distribution of "1's" to "6's" would be uniform (equally-likely). And, on any one roll, we expect uniform probability of "1's" to "6's."
David W.
Cam, from you thoughtful question and answer, I can tell that you are at least a thinker. Although I've studied lots of math and a bit of statistics, I'm a retired computer person, so I find that the math and stat tutors provide rather exact and terse answers (and I've posted errors and found several errors in others' posts).
I did write, "However, if you want to know the result at the end of multiple rolls, you have some math to do." That is where I can explain this so you can understand it.
Let's start with theoretical probability. The probability of any outcome is defined as the fraction or percent of successful outcomes over the total number of outcomes. So, if a coin in flipped once, the probability of heads is 50% and the probability of tails is 50%. They add up to certainty (P=1) because, we assume, the coin will not land and balance on it's edge (it helps me to have a sense of humor).
Now the probability of heads and tails on the second flip is, independently, 50-50. But you want to know what is the probability of two heads in a row. So, on the second flip the probability is 1/2, but half of those occur when the first flip resulted in heads and half of them occur when the first flip resulted in tails. Thus, the probability of getting two heads is 1/4.
We can also list all of the results: HH, HT, TH, TT and say P(HH)/total = 1/4
or use the formula (1/2)*(1/2) to get the probability of each result.
This would apply to your case if you asked, "What is the probability of getting 11 heads in a row?" That is, the probability of the same success if the individual probability of success is 1/2? Yes, mathematically, it is:
P = (1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)*(1/2)
P = 0.049%
This is why you will see lots of factorials and "to the power of" in statistics formulas.
And, it is smaller and smaller as you require a continuous string of successes. But -- theoretically -- it never quite reaches P=1 (certainty).
As, mentioned before, a success on an intermediate roll counts toward the success probability
only if all the rolls so far have been successful; otherwise this is one of the failures. That's where most people get confused.
Now, for two dice, this gets interesting, because the "success" might be "doubles." So, out of the 36 possible outcomes of rolling two dice, how often does "doubles" occur? 6 out of 36 = 1/6 = 16.6%. (note: with this problem, we don't care which die was first and which was second; we might have rolled them together).
Hey, now you a budding statistician!
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07/04/15
Cam R.
Great Answer - thank you... Happy 4th !
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07/04/15
Cam R.
07/04/15