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

[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.