Throwing Dice More Than OnceDate: 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/ |
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