On Average How Many Blackjack Hands Have to be Played to Win 5 in a Row

[James989]

I used the two online calculators to find the number 14, but the exact wording there is the average number of flips to have a 3 or more head in a row.

It should be exactly 3 head. Is it possible to have 20 head streak( more than 3 head) in 14 rounds ?

[aceside]

It should be exactly 3 head. Is it possible to have 20 head streak( more than 3 head) in 14 rounds ?

That is impossible in only 14 rounds. However, let me give you an example of having both 5 head and 3 head in a 24-round flip.
Let us define x=head, o=tail.

xox,xoo,oxx,oxo,xxx,xxo,oox,ooo

This represents a set of 8 3-round flips, and there is one 3-head flip, xxx; however, there is also one 5-head flip, xxxxx.
It all depends on how many flips we are going to do, there will be 3-head, 4-head, 5-head ... and so on.

[James989]

That is impossible in only 14 rounds. However, let me give you an example of having both 5 head and 3 head in a 24-round flip.
Let us define x=head, o=tail.

xox,xoo,oxx,oxo,xxx,xxo,oox,ooo

This represents a set of 8 3-round flips, and there is one 3-head flip, xxx; however, there is also one 5-head flip, xxxxx.
It all depends on how many flips we are going to do, there will be 3-head, 4-head, 5-head ... and so on.

The average 14 rounds required is based on below rules :-

1) You flip a fair(50/50) coin continuosly.
2) once you get 3 head in a row, you STOP immediately and record total no of flips since first flip, including the last three flips, say R.
3) You repeat 1) and 2) for X times.
4) Average rounds to have a 3 head in a row = (R1 + R2 + R3 + . . ...Rx)/X = 14. !

If involve tie/push(break or not break the streak), as per Midwest PlayerBJ promo, it will be more complex, you have to solve 5 simultaneous equations as describe in post #9. For Midwest Player BJ promo, the average rounds required to have a 5 winning streak(pushes not break the streak) is 84.28441, with p=0.4332, q=0.4788,r = 0.0880.

If p=0.4332, q=0.5668,r = 0, average rounds required to have a 5 winning streak is 113.88099.
If p=0.475, q=0.525,r = 0, average rounds required to have a 5 winning streak is 76.86738.

When r = 0, it does not matter pushes break the streak or not.

[aceside]

The average 14 rounds required is based on below rules :-

1) You flip a fair(50/50) coin continuosly.
2) once you get 3 head in a row, you STOP immediately and record total no of flips since first flip, including the last three flips, say R.
3) You repeat 1) and 2) for X times.
4) Average rounds to have a 3 head in a row = (R1 + R2 + R3 + . . ...Rx)/X = 14. !
.

This becomes clearer for me now. Can you demonstrate the calculation for the average number of required flips for a 3 head streak? It seems to me the above number X is infinite and thus it is impossible to calculate. If I can understand this number 14, I should be able to understand your blackjack streak calculator.

[Gronbog]

It seems to me the above number X is infinite and thus it is impossible to calculate

James989 has provided a methodology for computing the number by simulation. X is the number of times you run the simulation. You choose X. The higher X is, the more accurate the result will be (i.e. more correct decimal places).

[aceside]

James989 has provided a methodology for computing the number by simulation. X is the number of times you run the simulation. You choose X. The higher X is, the more accurate the result will be (i.e. more correct decimal places).

But this number 14 is a magic number. It may not need the recursive method of solving a set of three equations. My concern is still about if it is exactly a 3-head steak or a 3-or-more-head streak.

[DSchles]

But this number 14 is a magic number. It may not need the recursive method of solving a set of three equations. My concern is still about if it is exactly a 3-head steak or a 3-or-more-head streak.

It's exactly a three-head streak, because you stop as soon as you get three heads! Then you record that number. You do this over and over again and take the arithmetic average of the results. That number is 14. Stop beating this to death. There's nothing more to be said about it.

Don