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Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. Given, two six-sided dice are rolled. The possible outcomes 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) We have to find the probability of obtaining a 6 and a 6. P = ratio of the number of favorable outcomes to the total number of outcomes of an event ⇒ P = 1/36 This implies the probability of obtaining a 6 on the first die and a 6 on second die. From the possible outcomes, P(6 and 6) = 1/36 Therefore, P(6 and 6) = 1/36 Summary: The probability of rolling two six-sided dice and obtaining a 6 and a 6 is 1/36.
Rolling two six-sided dice is common in many of the most popular board games, including Monopoly, Backgammon and The Settlers of Catan. Even if you don't memorize the percentages in the chart below, it's helpful to have a general sense of how common each roll is. When it is your turn and you are two spaces away from landing on an opponent's hotel in Monopoly, this probability chart may comfort you. The chance of rolling a total of 2 is 2.78 percentThe chance of rolling a total of 3 is 5.56 percentThe chance of rolling a total of 4 is 8.33 percentThe chance of rolling a total of 5 is 11.11 percentThe chance of rolling a total of 6 is 13.89 percentThe chance of rolling a total of 7 is 16.67 percentThe chance of rolling a total of 8 is 13.89 percentThe chance of rolling a total of 9 is 11.11 percentThe chance of rolling a total of 10 is 8.33 percentThe chance of rolling a total of 11 is 5.56 percent The chance of rolling a total of 12 is 2.78 percent There's only one combination that yields a total of 2—when each die displays a 1. Likewise, there is only one combination that yields a total of 12—when each die displays a 6. They are the least likely combinations to occur. As you can see, 7 is the most common roll with two six-sided dice. You are six times more likely to roll a 7 than a 2 or a 12, which is a huge difference. You are twice as likely to roll a 7 as you are to roll a 4 or a 10. However, it's only 1.2 times more likely that you'll roll a 7 than a 6 or an 8. Another way of looking at these numbers is that, over time, you will roll one 4 or 10 for every two 7s rolled. You'll see six 7s for every 2 or 12. Of course, dice have a bad habit of defying expectations, so don't rely on probabilities from a chart like this to work out your game strategy. You're likely to find yourself on the losing end of a game if you do.
 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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You roll two six-sided dice what is the probability
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