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Die rolling probability

We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pants teal style avatar for user Brian Lipp
    why isn't the prob of rolling two doubles 1/36? prob of rolling any number on 1 dice is 1/6 shouldn't you multiply the prob of both dice like in the first coin flip video? I understand the explanation given, but I'm trying to figure out why the same coin logic doesn't work.
    (22 votes)
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    • leaf green style avatar for user Mrs. Signorello
      You need to consider how many ways you can roll two doubles, you can get 1,1 2,2 3,3 4,4 5,5 and 6,6 These are 6 possibilities out of 36 total outcomes. The probability for rolling one of these, like 6,6 for example is 1/36 but you want to include all ways of rolling doubles.
      (47 votes)
  • leaf blue style avatar for user Zain
    If this was in a exam, that way of working it out takes too long so is there any quick ways so you won't waste time?
    (17 votes)
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    • piceratops ultimate style avatar for user flyswatter
      well you can think of it like this. It really doesn't matter what you get on the first dice as long as the second dice equals the first. so the probability of the second equaling the first would be 1/6 because there are six combinations and only one of them equals the first.
      (29 votes)
  • piceratops ultimate style avatar for user Nusaybah
    At is there a mathematical reason why the favourable outcomes line up on the diagonal?
    (19 votes)
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    • leaf grey style avatar for user Qeeko
      That is a result of how he decided to visualize this. Imagine we flip the table around a little and put it into a coordinate system. Along the x-axis you put marks on the numbers 1, 2, 3, 4, 5, 6, and you do the same on the y-axis. We are interested in rolling doubles, i.e. getting the same on both dice. If we let x denote the number of eyes on the first die, and y do the same for the second die, we are interested in the case y = x. But this is the equation of the diagonal line you refer to.
      (15 votes)
  • aqualine seed style avatar for user Alisha
    At 2.30 Sal started filling in the outcomes of both die.
    This video wasn't what i was looking for but some of you might be able to help. I had a question:
    "Two dice are rolled, copy and complete the table below"......
    Then there was a table which looked exactly like the one Sal drew. But it had been filled out differently:
    In the first box (Dice 1 and Dice 2) it had been filled in as 2. Sal wrote 1,1. In another box on my sheet (1 across and three down on Sal's diagram) it had been filled out as 1. I am very confused on how they got this answer so if you understand what I'm talking about please answer. Sorry for the bad explaining!
    (13 votes)
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  • orange juice squid orange style avatar for user Gabrielle
    Is there a way to find the probability of an outcome without making a chart?
    (6 votes)
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    • blobby green style avatar for user Baker
      Probably the easiest way to think about this would be:
      P(Rolling a 1 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296
      P(Rolling a 2 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296
      P(Rolling a 3 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296
      P(Rolling a 4 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296
      P(Rolling a 5 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296
      P(Rolling a 6 four times in a row) = 1/6 * 1/6 * 1/6 * 1/6 = 1/1296

      Adding these probabilities together, we get: 6/1296 = 1/216.

      So the probability you will roll the same number four times in a row with a fair dice is 1/216.
      Does this help.
      (9 votes)
  • old spice man green style avatar for user Cal
    I was wondering if there is another way of solving the dice-rolling probability and coin flipping problems without constructing a diagram? Is there a way to find the solution algorithmically or algebraically?
    (3 votes)
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    • female robot grace style avatar for user loumast17
      Definitely, and you should eventually get to videos descriving it. But to show you, I will try and descrive how to do it.

      So, since there is an equal chance to roll any number on a six sided die, that means the chance of rolling any one number is one out of 6 or 1/6. You can see that with a diagram. Now, rolling two different numbers in a specific order you can tell with a diagram is 1/36. To find this out through math though you multiply probabilities of events happening if you are looking for both of them happening. so you want to roll x first AND y second. so that's 1/6*1/6, which is 1/36. And you can keep goign with this pattern.

      The calculation is a bit different if you are looking for one thing to happen OR another. In simple caes it's just adding, like what are the odds of rolling a 1 OR 2 on a dice? you add the two, which you can see on a diagram.

      You still have to be careful, like if a problem asks what the odds of rolling a 1 AND a 2 on ONE die is, you can't roll both so the answer is 0. Or rolling a 1 on one die OR rolling a 2 on another. It's still 1/6 since you are rolling them separately and they don't effect each other. It can get a bit confucing, but most of the time you will be using the more simple cases. If ever in doubt use diagrams to see a few cases of what you're doing to get an intuition
      (8 votes)
  • blobby green style avatar for user Kratika Singh
    Find the probablility of the occurance of1on a die if it has one more of its faces marked as 1instead of 6
    (3 votes)
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    • male robot donald style avatar for user Admiral Betasin
      Here's how you'd do the problem.
      We know that two of the sides have 1. The rest have other numbers.
      Since in total, there are 6 sides, get the amount of sides that have 1 divided by 6 (six possible outcomes) and that's your answer.
      So in your case, it's 2/6, which is 1/3.
      (8 votes)
  • duskpin tree style avatar for user Errol
    Can learners open up a black board like Sals some where and work on that instead of the space in between problems?
    (4 votes)
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    • duskpin ultimate style avatar for user BeeGee
      If you're working on a Windows pc, you would need either a touchscreen pc, complete with a stylus pen or a drawing tablet. Then you could download for free the Sketchbook Pro software for Windows and invert the colors. Voila, you have a Khan Academy style blackboard.
      If you don't have a pc with a touchscreen or drawing tablet, you can use phones with drawing styluses like the Samsung Note phones to do your work on SketchBook mobile.
      Or just, you know, do it on plain lined, graphing, or print paper (^ u^
      hope this helped.
      (4 votes)
  • aqualine ultimate style avatar for user alyxi.raniada
    Can someone help me
    "If you roll two fair six-sided dice, what is the probability that at least one die shows a 3?"
    (3 votes)
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  • blobby green style avatar for user Sukhman Singh
    From a well shuffled 52 card's and black are removed from cards find the probability of drawing a king or queen or a red card
    (3 votes)
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Video transcript

Find the probability of rolling doubles on two six-sided dice numbered from 1 to 6. So when they're talking about rolling doubles, they're just saying, if I roll the two dice, I get the same number on the top of both. So, for example, a 1 and a 1, that's doubles. A 2 and a 2, that is doubles. A 3 and a 3, a 4 and a 4, a 5 and a 5, a 6 and a 6, all of those are instances of doubles. So the event in question is rolling doubles on two six-sided dice numbered from 1 to 6. So let's think about all of the possible outcomes. Or another way to think about it, let's think about the sample space here. So what can we roll on the first die. So let me write this as die number 1. What are the possible rolls? Well, they're numbered from 1 to 6. It's a six-sided die, so I can get a 1, a 2, a 3, a 4, a 5, or a 6. Now let's think about the second die, so die number 2. Well, exact same thing. I could get a 1, a 2, a 3, a 4, a 5, or a 6. Now, given these possible outcomes for each of the die, we can now think of the outcomes for both die. So, for example, in this-- let me draw a grid here just to make it a little bit neater. So let me draw a line there and then a line right over there. Let me draw actually several of these, just so that we could really do this a little bit clearer. So let me draw a full grid. All right. And then let me draw the vertical lines, only a few more left. There we go. Now, all of this top row, these are the outcomes where I roll a 1 on the first die. So I roll a 1 on the first die. These are all of those outcomes. And this would be I run a 1 on the second die, but I'll fill that in later. These are all of the outcomes where I roll a 2 on the first die. This is where I roll a 3 on the first die. 4-- I think you get the idea-- on the first die. And then a 5 on the first to die. And then finally, this last row is all the outcomes where I roll a 6 on the first die. Now, we can go through the columns, and this first column is where we roll a 1 on the second die. This is where we roll a 2 on the second die. So let's draw that out, write it out, and fill in the chart. Here's where we roll a 3 on the second die. This is a comma that I'm doing between the two numbers. Here is where we have a 4. And then here is where we roll a 5 on the second die, just filling this in. This last column is where we roll a 6 on the second die. Now, every one of these represents a possible outcome. This outcome is where we roll a 1 on the first die and a 1 on the second die. This outcome is where we roll a 3 on the first die, a 2 on the second die. This outcome is where we roll a 4 on the first die and a 5 on the second die. And you can see here, there are 36 possible outcomes, 6 times 6 possible outcomes. Now, with this out of the way, how many of these outcomes satisfy our criteria of rolling doubles on two six-sided dice? How many of these outcomes are essentially described by our event? Well, we see them right here. Doubles, well, that's rolling a 1 and 1, that's a 2 and a 2, a 3 and a 3, a 4 and a 4, a 5 and a 5, and a 6 and a 6. So we have 1, 2, 3, 4, 5, 6 events satisfy this event, or are the outcomes that are consistent with this event. Now given that, let's answer our question. What is the probability of rolling doubles on two six-sided die numbered from 1 to 6? Well, the probability is going to be equal to the number of outcomes that satisfy our criteria, or the number of outcomes for this event, which are 6-- we just figured that out-- over the total-- I want to do that pink color-- number of outcomes, over the size of our sample space. So this right over here, we have 36 total outcomes. So we have 36 outcomes, and if you simplify this, 6/36 is the same thing as 1/6. So the probability of rolling doubles on two six-sided dice numbered from 1 to 6 is 1/6.