Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? Show SolutionWe split this problem into cases.First, we calculate the probability that all four are the same. After the first dice, all the numbers must be equal to that roll, giving a probability of .Second, we calculate the probability that three are the same and one is different. After the first dice, the next two must be equal and the third different. There are orders to roll the different dice, giving .Adding these up, we get , or .Solution 2Note that there are two cases for this problem : Exactly three of the dices show the same value.There are values that the remaining die can take on, and there are ways to choose the die. There are ways that this can happen. Hence, ways. : Exactly four of the dices show the same value.This can happen in ways.Hence, the probability is Solution 3We solve using PIE. We first calculate the number of ways that we can have dice be the same and the other dice be anything. We therefore have ways to have at least dice be the same.But wait! We have overcounted the case where all dice are the same! Since the previous case occurs in each of these cases times, we must subtract the -dice total three times in order to have them counted once. There are ways to have four dice be the same, so we our total count is .Therefore, our probability is , which is answer choice .-FIREDRAGONMATH16 Solution 4There are two cases to consider: Three of the dice roll the same number, and all four of the dice roll the same number.
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. What is the probability of rolling the same number 3 times on a dice?The three dice are rolled fairly without any cheating. Each of the dice rolls is an Independent Event, that is the outcome from anyone dice roll has no impact whatsoever on the outcome of any other dice roll. The probability of all three happening is the product of the three probabilities: 1 × (1/6) × (1/6) = 1/36.
What is the probability of rolling a 3 if you rolled a 6 sided dice?Originally Answered: If you roll a six sided dice, what is the probability of getting 3? There are six numbers possible. The probability of getting the number 3 out of 1,2,3,4,5,6 is 1/6.
What is the probability of rolling a 5 on a 6 sided dice?There is one side marked 5 on a standard six sided die. Assuming we are talking such a die, and that the die is honest, then the odds are 1/6. However, there's a die which is marked all 5s, useful in allowing you to always roll a 7 or 11 when combined with a second special die marked with half 2s and half 6s.
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