exam probability question

Question

Consider the probability experiment of a sequence of independent trials until a success occurs, each trial is the rolling a pair of 6-sided dice. Success is the occurrence of the random event A. Suppose the event A is “at least one roll is equal to 4” and a random variable is the number of trials required.
Find the probability of the success for a single trial, find and plot a graph of probability mass function of the random variable , find a mathematical expectation and variance of . Perform the probability experiment N times (N = 60). Represent the results in table form. Estimate the mathematical expectation of using the experimental results.

Answer 1

Jason's method will simulate trials, but it won't give you what (I think) the problem is asking of you. The problem is disentangling the experimental *design* from the desired *data*. You have to track a couple of items, to get everything you need. Now, you might choose to solve this problem as an infinite-number-of-trials problem (as implied by "probability mass function"), or as a discrete-number-of-trials problem.The issue here is that "perform the probability experiment" is a bit vague here -- have you rolled 60 times, or have you rolled until you had a success, 60 times? They're quite different!

I'll solve for the general case first below, then sketch what to do for a discrete-rolls result.

Looks to me (the mathematical terminology is a bit dense!) like the problem is anticipating that you stop rolling the dice once you get a 4. The input datum is then the number of times you have rolled to get to that result.,

Myself, I'd prefer to set up the initial probability in terms of NOT ROLLING a 4. That's (5/6)*(5/6) = 25/36. As Jason notes, that's equivalent to a 11/36 probability of rolling at least one 4 (per trial).

So for each iteration down the table, one desired item to track is the probability of getting to that row and stopping there. For the third row, for example, that would be (25/36)*(25/36)*(11/36), because you passed through the first two rolls NOT ROLLING a 4, then did roll a 4. I'll call this value (for each row) the incremental probability (of "landing" there).

The incremental_value (for the desired statistic, number of trials required) of getting to that row is then the probability of getting there, times the row number (you've rolled the number of times equal to the row number). So for row 1, it's (11/36)*1; for row 2, (25/36)*(11/36)*2; for row 3, (25/36)^2 * (11/36) * 3, and so on.You may now (after 60 rows, for example -- that is, you rolled 60 times or until you had a "hit", for each trial) assess the data of this probability function. The overall probability of having gotten a 4 is the sum of the incremental probabilities, but the overall expected number of rolls required is the sum of the incremental_values (well, actually, the residual probability of failure *60 should also be added). The incremental probabilities (spread over the domain of 1 - 60 trials) are the appropriate data to analyse for variance of the number of rolls required (again, with the last incremental probability adjusted, since the problem didn't say "if you succeed, how many rolls....").Now, if you consider this as a simulation problem, there are two ways to go about it. You could "roll" the dice in a loop program and report back out the number of rolls required (the inefficient way!), or you could go about doing it mathematically (efficiently!). To do the latter, first, partition your domain into how ever many "maximum rolls" you want to perform -- let's say 100. You have to assign a cut-off, otherwise you might go on rolling forever! Next, examine the properties of the "incremental probability" function above. Generate two random numbers, the first scaled to run as integers from 1 to your maximum number of rolls. Suggestion = int(RND*maxrolls + 1) . Keep track of this number; it is the number of rolls you have performed IF you accept the value. Next, compare the second random number, scaled up so as to include the range (0,max_incremental probability), to the value of the incremental probability at that domain point. If the scaled second random number is <= the incremental probability, then accept the domain value, and add it as a new member to a set of accepted values. Keep track of how many values you have accepted. When you have your target number of data, stop generating data and analyse them in conventional manner.This may appear (and be) inefficient for a probability distribution that "falls off" abruptly, but it handles all sorts of distributions nicely WITHOUT the problems of programming multiple dice rolls.Incidentally, you should be calculating the incremental probabilities as a set only once on the spreadsheet, and comparing to the appropriate cell of these values when you do the test for acceptance of each prospective datum. If you have some Excel programming experience, you might recognize the utility of doing the data generation within a "macro" program on your spreadsheet -- that way, you can input the domain limit and the number of target data and (F5) or whatever starts your macro. Maybe even set up as a displayed graph to show your accumulating data!I have no idea what "experimental results" are being referred to as a "mathematical expectation of using". Perhaps they mean the number of accepted data, divided by the total number of rolls performed. Perhaps not!

Cheers, -- S. de Riel

Answer 2

To first solve this, we need to figure out the probability that at least one roll is a 4, which means either one roll is a 4 or both are. You can use the binomial distribution formula to solve for that.

 

P(1 roll is a 4) = 2C1 * (1/6) * (5/6) = 2 * (5/36) = 10/36

 

P(both rolls are a 4) = 2C2 * (1/6) * (1/6) = 1/36

 

P(at least one roll is a 4) = 10/36 + 1/36 = 11/36

 

If you wanted to simulate the results of the dice rolls n=60 times, then you are going to be best off using the random numbers function on Excel (or if your school is teaching you any statistical programs like R or MATLAB, then you should use that). You basically can pick a random number between 1-6 twice (representing the two die) and then define it as a success if at least one is randomly chosen to be a 4. Then repeat this 60 times and analyze the result of successes vs failures.

 

 

Edit: Stanton is correct and gave a great explanation. This is what's known as a random walk, as you don't know how many times it will take to reach your destination for your random variable (in this case, rolling a 4).

 

 

To punish myself for misreading the question, I tried coding the solution myself. It's actually pretty simple if you're familiar with how to use while loops and if/else logic.

 

 

Here is a histogram of the results of the random walk to give you an idea of what the output should be (this is the number of turns it took to finally roll at least one 4):

 

 

https://s28.postimg.org/io90105i5/rj1.png

 

 

My results had an average of 3.7 rolls to get at least one 4, which is skewed by the couple of outlier results where is took more than 10 rolls. Based on the probability we calculated, the number should normalize to about 3.27 over time (3.27 is 36/11, which is the ratio of total rolls to rolls with no 4s to rolls with at least one).