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Neither method is correct.
In order to use the formula $$ \frac{\text{favorable outcomes}}{\text{total outcomes}}, $$ you need your events to be equally likely. In the first method, the outcomes $(1,1)$ and $(1,2)$ are not equally likely. The first method requires both dice to have a value $1$ while the second method has two situations for the dice. To make this clearer, suppose that the dice are red and blue. Then, $(1,1)$ means that both the red die and the blue die show $1$. On the other hand, in the first method, $(1,2)$ represents the two possibilities ($1$-Red and $2$-Blue) or ($2$-Red and $1$-Blue). Since there are two possible ways to get a $1$ and a $2$, this $(1,2)$ has double the chances of occurring when compared to $(1,1)$. For the second formulation, you're double counting the pairs of the form $(1,1)$. In this case, you're trying to describe $(1,1)$ for $1$-Red and $1$-Blue as well as $(1,1)$ for $1$-Blue and $1$-Red, but these are exactly the same situation. Therefore, in the second case, you shouldn't duplicate the pairs that are identical under reversing the coordinates. To calculate the probability correctly, the list should be $$ (1,1),(1,2),(1,3),(2,1),(2,2),(3,1). $$ Or, in other words, for the red and blue dice, $$ (1R,1B),(1R,2B),(1R,3B),(2R,1B),(2R,2B),(3R,1B). $$ Since there are $6$ possibilities for the red die and $6$ possibilities for the blue die, this results in $36$ total possible outcomes. Putting this all together, the probability is $6/36=1/6$.
Contents: Watch the video for three examples: Probability: Dice Rolling Examples Watch this video on YouTube. Can’t see the video? Click here. Need help with a homework question? Check out our tutoring page! Dice roll probability: 6 Sided Dice ExampleIt’s very common to find questions about dice rolling in probability and statistics. You might be asked the probability of rolling a variety of results for a 6 Sided Dice: five and a seven, a double twelve, or a double-six. While you *could* technically use a formula or two (like a combinations formula), you really have to understand each number that goes into the formula; and that’s not always simple. By far the easiest (visual) way to solve these types of problems (ones that involve finding the probability of rolling a certain combination or set of numbers) is by writing out a sample space. Dice Roll Probability for 6 Sided Dice: Sample SpacesA sample space is just the set of all possible results. In simple terms, you have to figure out every possibility for what might happen. With dice rolling, your sample space is going to be every possible dice roll. Example question: What is the probability of rolling a 4 or 7 for two 6 sided dice? In order to know what the odds are of rolling a 4 or a 7 from a set of two dice, you first need to find out all the possible combinations. You could roll a double one [1][1], or a one and a two [1][2]. In fact, there are 36 possible combinations. Dice Rolling Probability: StepsStep 1: Write out your sample space (i.e. all of the possible results). For two dice, the 36 different possibilities are: [1][1], [1][2], [1][3], [1][4], [1][5], [1][6], [2][1], [2][2], [2][3], [2][4], [2][5], [2][6], [3][1], [3][2], [3][3], [3][4], [3][5], [3][6], [4][1], [4][2], [4][3], [4][4], [4][5], [4][6], [5][1], [5][2], [5][3], [5][4], [5][5], [5][6], [6][1], [6][2], [6][3], [6][4], [6][5], [6][6]. Step 2: Look at your sample space and find how many add up to 4 or 7 (because we’re looking for the probability of rolling one of those numbers). The rolls that add up to 4 or 7 are in bold: [1][1], [1][2], [1][3], [1][4], [1][5], [1][6], There are 9 possible combinations. Step 3: Take the answer from step 2, and divide it by the size of your total sample space from step 1. What I mean by the “size of your sample space” is just all of the possible combinations you listed. In this case, Step 1 had 36 possibilities, so: 9 / 36 = .25 You’re done! Two (6-sided) dice roll probability tableThe following table shows the probabilities for rolling a certain number with a two-dice roll. If you want the probabilities of rolling a set of numbers (e.g. a 4 and 7, or 5 and 6), add the probabilities from the table together. For example, if you wanted to know the probability of rolling a 4, or a 7:
Probability of rolling a certain number or less for two 6-sided dice.
Dice Roll Probability TablesContents: Probability of a certain number with a Single Die.
Probability of rolling a certain number or less with one die.
Probability of rolling less than certain number with one die.
Probability of rolling a certain number or more.
Probability of rolling more than a certain number (e.g. roll more than a 5).
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Visit out our statistics YouTube channel for hundreds of probability and statistics help videos! ReferencesDodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.
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