Really basic probability question

[Morbo]

This is surely most elementary, but perhaps I fell asleep during this mathclass. Let's imagine we are going to roll dice to hit a six. The odds of hitting a six with one roll is 1/6, or 16%. If we were to roll two dices, the odds would be 2/6, ro 33%. If we roll three dices, we are up to 50%. Four dices gives us 67% to score a six and five dices gives us 83%.

Now here's what I don't understand. What are the odds of scoring a six when you roll six dices? Math says it should be 100% but alas it could not be and are not possible. Where is my math flawed?

Now I do realize that if I'd roll six dices one million times I would score a six about one million times aswell, but I want to know the odds of scoring a six on a single six-diced roll...

[BruceZ]

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Let's imagine we are going to roll dice to hit a six. The odds of hitting a six with one roll is 1/6, or 16%.

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True, for 1 die.

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If we were to roll two dices, the odds would be 2/6, ro 33%.

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Not true since this double counts the times you get 2 sixes, which happens 1/36 of the time. If you subtract this, the probability becomes 2/6 - 1/36 = 11/36 = 30.6%. Out of 36 possible outcomes, only 11 contain a 6.

You can also compute this as 1 minus the probability of not getting a six on either die. The probability of not getting a six is 5/6 for each die, so the probability of not getting it on either die is (5/6)^2 since the dice are independent. The probability that you do get a six is 1 - (5/6)^2 = 11/36 = 30.6% as before.

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If we roll three dices, we are up to 50%. Four dices gives us 67% to score a six and five dices gives us 83%.

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For three dice it is 1 - (5/6)^3 = 42.1%. Four dice gives 1 - (5/6)^4 = 51.8%. Five dice gives 1 - (5/6)^5 = 59.8%.

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Now here's what I don't understand. What are the odds of scoring a six when you roll six dices? Math says it should be 100% but alas it could not be and are not possible. Where is my math flawed?

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Math says it should be 1 - (5/6)^6 = 66.5%.

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Now I do realize that if I'd roll six dices one million times I would score a six about one million times as well

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If you roll 6 dice a million times, you would score at least 1 six on an average of 665,102 rolls. You would roll an average of 1 million sixes all together, if you count double sixes twice.

[Morbo]

I semi-understand. What happens if we change the rules so that the goal is to hit any amount of sixes, but atleast one?

[BruceZ]

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I semi-understand. What happens if we change the rules so that the goal is to hit any amount of sixes, but atleast one?

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I assumed that was the goal. I computed the probability of hitting at least one six throwing N dice, and that probability is 1 - (5/6)^N. That is 1 minus the probability of having no sixes on all N dice. What was the rule before? What didn't you understand?

You were actually computing the average number of sixes because you were counting every six. The average number of sixes with N dice is N/6. That is the expected value of the number of sixes, and that is different from the probability of throwing at least 1 six.

[Morbo]

Oh, now I get it /images/graemlins/blush.gif /images/graemlins/grin.gif