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CCSS.Math: pascale rickets has invented a game called three rolls to ten you roll a fair six-sided die three times if the sum of the rolls is 10 or greater you win if it is less than ten you lose what is the probability of winning three rolls to ten so there are several ways that you can approach this the way we're going to tackle it in this video is we're going to try to come up with an experimental probability we're going to do many experiments trying to win three rolls to ten and figure out the proportion that we actually win and the more experiments we try the better the more likely that we're going to get a good approximation of the actual probability so let's do that and to help us I'm going to have a computer generate a string of random digits from 0 to 9 and the way that we're going to use this is remember we're rolling a fair six-sided die so the outcome could be 1 2 3 4 5 or 6 for each roll in this random number list that the computer has generated I do get digits from 1 to 6 but I also get the digits 7 8 9 and 0 and so what I'm going to do for each experiment I'm going to start at the top left and I'm going to consider each digit a roll if it gives me an invalid result for a six-sided die so if it's a 0 + 8 a 7 or 9 I will just ignore that I will just say well that wasn't a valid roll it's like you roll the die and it fell off the table or something like that so let's do that let's do multiple experiments of taking three rolls sum them up and we'll see how many we can do to figure out an experimental probability of winning Pascal's game so let me set up a little table here so I want space to show the sum so this is going to be the experiment experiment so let me write the sum and over here we're going to say did we win all right so let's start with experiment one so our first roll we got a one our second roll we got a 5 we're doing quite well and then our third roll we got a six did we win well one plus five plus six is 12 yes we won let's do another experiment this is going to be experiment two we can just keep going here these are random digits so we have a six in our first roll we got a two in our second roll we got a four in our third roll did we win yes once again this sums up to twelve so we won all right let's do another experiment so experiment number three so this first thing is invalid so this is our first roll we got a six and then this is invalid our second roll we get a three this is invalid that is invalid that is invalid and then in our third roll we got a two so we squeaked by this adds up to eleven yes that looks like a win all right let's do our fourth experiment here so our first roll we got a 1 this is invalid second roll we got a 2 this is invalid third roll we get a 5 did we win one plus two plus five is eight no we did not win so that's our first non win so let's keep going this is interesting all right this is invalid so we're going to have so this is trial 5 we are going to have 4 plus 3 plus 1 4 plus 3 plus 1 adds up to 8 did we win no let's just keep going here so I'm going to keep going with my table where I have experiment I'll do 5 more trials X bar and some and do we win let me make the table this is just a continuation of the table we had before I don't want to go below the page because I want to be able to look at our random numbers here so we are on to experiment six experiment six we are getting a three in the first roll a three in the second roll this isn't looking good and then a two in our third roll did we win no this is less than 10 now we go to experiment seven experiment seven we get a two in our first roll this is invalid we get a three in our second roll plus 3 and we get a 1 in our third roll so plus one once again we did not win now we go to experiment we will go to experiment 8 we get a 1 in our first roll we get a 3 in our second roll this is invalid the die fell off the table we can think of it that way and then in our third roll we get a 5 plus 5 did we win no this adds up to 9 so we had a string of wins to begin with but now we're getting a string of non wins all right now let's go to experiment 9 so we get a 6 in our first roll we get a 4 in our second roll and then these are all invalid and then we get a 5 in our third roll did we win here yes we won over here this is definitely going to be greater than 10 this is 15 here alright last experiment or at least for this video last experiment you could keep going in fact I encourage you to after this to see if you can get a more accurate a better approximation of the theoretical probability of winning by doing more experiments to calculate an experimental probability so here your experiment 10 first roll we get a 5 second roll we get a 2 this is invalid invalid invalid then we get a 6 here we definitely won so with 10 trials based on 10 trials or 10 experiments what is our experimental probability of winning this game well out of the 10 experiments how many did we win it looks like we won 1 2 3 4 5 so based on just these 10 experiments we've got a pretty clean 50% so do you think the theoretical probability is actually 50% maybe you'd want to continue running these experiments over and over maybe we'd want to do a computer program that could run this experiment set of 10 times maybe 10,000 times to see if we can get closer to the true theoretical probability

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