Throwing Dice More Than Once

Date: 09/25/97 at 16:11:50
From: Vicken Merjanian
Subject: Probability

If I throw two dice once, the probability of getting sum = 2 is 1/36.
The probability of getting sum = 3 is 2/36. So far I understand this 
concept, but I don't understand what happens if I throw the dice more 
than once. If I throw one die once the probability of getting any one 
of the six numbers is 1/6. Does this change if I throw more than once?

I tried to solve it by assuming I threw a die 100 times; my
probability increases by 100/100 which = one. So my probability stays
the same, but that doesn't make sense since it should increase with 
more throws, right?

Could you please explain the logic behind this?
Thanks so much.

Date: 12/02/97 at 14:25:05
From: Doctor Sonya
Subject: Re: Probability

Dear Vicken,

I'm not sure I understand your question, so If I haven't interpreted 
it right, please write me back. 

I'll start from the beginning.  Everything in your first paragraph is 
right. You want to know if the probability of having any one of the 
six faces come up when you throw a die changes depending on how many 
times you throw it. The answer is no. No matter how many times you 
roll a die, the probability that a 3 will come up on a certain roll is 
ALWAYS 1/6.  This is because every roll of the die is exactly the 
same. The rolls that came before do not change the rolls that will 
come in the future. Said another way, the outcome of each roll has 
nothing to do with the outcomes of the other rolls. In probability 
language, the rolls are said to be independent.

I'm not quite sure what you are asking in your second paragraph. Are 
you talking about the number you get with one die or the sum of the 
numbers that come up when you roll two dice?  

If you are talking about the number that comes up when you roll one 
die, remember that because the rolls of the die are independent, even 
if you roll it 100 times, the probability of getting, say, 4, is still 
1/6.

If you are talking about the sum of the two numbers that come up when 
you roll two dice, the probability of getting a certain sum does 
change as the number of throws increases. If we think about this for a 
second, you'll see why it's true.

You correctly figured out that if you have two dice, and you roll them 
both once, the probability of their sum being 2 is 1/36. But how did 
you figure that out? There are 36 possible ways we can roll the two 
dice. (Do you see why this is true? There are 6 choices for the first 
die and 6 choices for the second die, and so there are 36 choices in 
all.)  There is only one way that the sum of the two rolls is 2, when 
both dice show a 1.  Thus the probability that the sum is two is 1/36.

Now let's say that we have three dice, and we roll them all once.  
What is the probability that the sum of these three rolls is 2? If you 
said zero, you're right. There is no way the sum can be 2 (do you see 
why?), so the probability of that happening is zero. As you may have 
noticed, this is different from the probability that 2 will be the sum 
of rolling TWO dice instead of three.

You also figured out that the probability of getting a total of three 
when you roll two dice is 2/36. What's the probability of getting 
three when we roll three dice?  Let's work out the problem together.

There are 6x6x6 possible ways we can roll the dice. There is only one 
way the sum of the numbers that come up can be three. Make sure you 
know when this is. Thus the probability that the sum equals three is 
1/(6x6x6), which is 1/216. This is again very different from the 2/36 
that you calculated for two dice. 

I hope this helps answer some of your questions about probability.  
Write back if there was anything that was unclear, or if you have any 
more questions.

-Doctor Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/