How many times will a seven be thrown in a roll using 2 dice and does the probability decrease after one or more are thrown.

Question

Ok we know that when 2 dice are thrown that a seven is the most dominant number and on average should appear 6 times in 36 rolls or 1 in 6 on average. My question has to do with the 7 being thrown in secession. Using the math above showing the 7 showing up 6 times in 36 rolls or 1 time in 6 on average does that mean that for the 7 to show 2 times in a row it would be 1 time in 36 rolls on average which I got to by multiplying 6x6. I am not sure if this is how you get to the answer I am looking for but if my math is correct would that mean for 3 sevens in a roll it should show up once in 216 rows and 4 times in a row once in 1,296 rolls on average. Through my experimenting I do notice that the 7 shows up on 6 times in 36 rolls or 1 in 6 on average but 2 times in a row which should be once in 36 rolls seems to slow down to about 50 or so rolls and for 3 sevens in a roll to show which mathematically should be 216 rolls that it seems to take longer. Does it have to do with probability of something happening in secession lesson as it increases or do I have my math completely wrong? Thank you. Scott in St.Pete Forida 

Answer 1

Your math is correct.

 

You have computed the Theoretical Probability.  This is the most likely probability (that is, the number of successful events divided by the total number of events) with "fair" dice (that is, the numbers 1-6 are equally likely to occur).  So, given an equal distribution of the events, you may compute the probability of three events happening in succession.

 

However, the Experimental Probability is what you cite.  You used the words, "Through my experimenting," "seems to slow down," and "seems to take longer."  So, what you see is not what you would expect.  That's because it is still possible, but less likely.  For example, it is possible, though not very likely, to flip a "fair" coin five time and get all five heads.  That doesn't change the Theoretical Probability, but it does make us re-validate whether the coin is "fair."

 

As an example:  Computer programs often simulate games.  They use pseudo-random number generators (the numbers are close to random, but not totally random).  Some pseudo-random number generators begin with a "seed" value (that is, a starting value).  Using the same seed, the pseudo-random number generator will generate the same sequence of pseudo-random numbers every time.  This may be useful for testing the logic of the program, but it makes the game quite predictable.  Thus, pseudo-random generators usually have a "randomize" option to get a new seed before generating numbers in a sequence (much like we demand that a deck of cards be shuffled).

 

You have experienced slightly unexpected (but certainly not impossible) events.

Answer 2

I see that you have used the word secession; I trust that you meant succession.

 

P(2 dice sum rolled is 7) is indeed 1/6. Each roll is independent of the prior roll.

 

P(total 7 twice in a row) is therefore (1/6)(1/6) = 1/36.

 

If you wished to determine probability of two rolls of total 7 each, in 3 rolls, then you have to figure that the non-7 total could be on any one of the first, second or third rolls. So that this probability computes as 3(1/6)(1/6)(5/6), I believe.