Die rolling probability

find the probability of rolling doubles on Tucson 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 2 and a 2 that is doubles a 3 and a 3 a 4 and a 4 or 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 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 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 in this let me draw a grid here just to make it a little bit neater so we draw a line there and then a line right over there let me draw actually several of these just so that we can really do this a little bit clearer so let me draw is let me draw a full grid all right and then let me draw the vertical lines vertical lines only a few more left there we go now all of this 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 the outcome for our role of 2 on the first die a 2 on the first die this is where I roll 3 on the first die 3 on the first i4 i think you get the idea on the first die and then a five on the first die five and then finally these are this row this last row is all the outcomes where our role is six on the first to die now we can go through the columns and this first column is where we roll a 1 on the second die a 1 on the second die this is where we roll a 2 on the second die so let's draw that out and write it out fill in the chart here's where we roll a 3 on the second die 3 3 this is a comma that I'm doing between the two numbers here's where we have a 4 here's where we have a 4 a 4 and then here's where we roll a 5 on the second die just filling this in filling this in almost this last column is where we roll a 6 on the second die 6 on the second die 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 no 1 on the second die this outcome is where we roll a 3 on the first die 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 six times six possible outcomes now with this out of the way how many how many of these outcomes satisfy our criteria satisfy the 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 a 1 it's a 2 and a 2 a 3 and a 3 of 4 and a 4 a 5 and a 5 and a 6 and a 6 so we have one two three four five six events satisfy this event are or are the outcomes that are consistent with this event now given that what it 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 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 figure that out over the total I want to do that pink color over the total number of outcomes over the size of our sample space so this right over here we have 36 total outcomes so we have 36 36 outcomes and if you simplify this 6 over 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