# Number of Possible First Roll Outcomes in Bowling **URL:** \ **Category:** The Game Room\ **Created:** [April 23, 2013, 7:04pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523 "2013-04-23T19:04:52Z")\ **Posts on this page:** 15\ **Page:** 1
**Author:** ![HeyHomie](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/heyhomie/32/207_2.png) [@HeyHomie](https://boards.straightdope.com/u/HeyHomie)\ **Post date:** [April 23, 2013, 7:04pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/1 "2013-04-23T19:04:52Z")
How many possible outcomes are there after rolling the first roll in bowling? The mathematical answer would be 10^9 (I think-- math isn’t my strong suit). But the mathematical answer would include outcomes that aren’t possible (e.g., only the 5, 8 or 9 pin falls), or are theoretically possible but practically impossible (e.g., only the 2 pin or 3 pin falls). Has anyone ever crunched the numbers and come up with a definitive number of possible outcomes? ---
**Author:** ![BorgHunter](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@BorgHunter](https://boards.straightdope.com/u/BorgHunter)\ **Post date:** [April 23, 2013, 7:16pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/2 "2013-04-23T19:16:22Z")
> [@HeyHomie](#): > > The mathematical answer would be 10^9 (I think-- math isn’t my strong suit). 2¹⁰ = 1024, actually. After the first roll, each pin has two possible states (standing or toppled) and there are ten of them. ---
**Author:** ![Agent\_Foxtrot](https://avatars.discourse-cdn.com/v4/letter/a/9fc29f/32.png) [@Agent\_Foxtrot](https://boards.straightdope.com/u/Agent_Foxtrot)\ **Post date:** [April 23, 2013, 8:42pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/3 "2013-04-23T20:42:58Z")
> [@BorgHunter](#): > > 2¹⁰ = 1024, actually. After the first roll, each pin has two possible states (standing or toppled) and there are ten of them. How would this be calculated for a standard 10-round game, since you skip a frame if you hit a strike? ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 23, 2013, 8:50pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/4 "2013-04-23T20:50:08Z")
!?!? Ok, I get it, different pin combinations. I thought you guys were nuts when the answer wasn’t 10. ---
**Author:** ![sevenwood](https://avatars.discourse-cdn.com/v4/letter/s/6de8d8/32.png) [@sevenwood](https://boards.straightdope.com/u/sevenwood)\ **Post date:** [April 23, 2013, 9:15pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/5 "2013-04-23T21:15:50Z")
> [@HeyHomie](#): > > But the mathematical answer would include outcomes that aren’t possible (e.g., only the 5, 8 or 9 pin falls) I’ve seen pin combinations left standing after the first ball that I’d have sworn were mathematically impossible. But I’ve seen a bowling ball hit a pin which in turn hits a second pin with an impact that _ **keeps the first pin from falling down** _. I’ve also seen a hit pin (hit by either the ball or another pin) _ **slide without falling** _ into a different pin position. (I believe that technically in the second case a pin that stays upright but slides out of position is supposed to be placed (by hand) back into its original pin position, but I’ve never seen anyone ask the house staff to do so.) ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 23, 2013, 9:18pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/6 "2013-04-23T21:18:32Z")
In Duck Pins the ball can pass between pins, so I think any combination is possible after the first roll. In Ten Pins it’s also possible to have any combination. Normally you can’t knock down the 5, 8, or 9 pin without hitting one of the other pins. But you can hit one of those other pins and move it without knocking it down. Hit just right it can slide and knock down one of those hidden pins. That’s very unlikely, but I’ve seen a sliding 2 knock over a 3 and stay up. Pins hit hard enough can get knocked down and bounce back up again if they interact with other pins also. So there are some incredibly unlikely combinations that could happen. eta: pretty much what sevenwood said ---
**Author:** ![BorgHunter](https://avatars.discourse-cdn.com/v4/letter/b/7c8e57/32.png) [@BorgHunter](https://boards.straightdope.com/u/BorgHunter)\ **Post date:** [April 24, 2013, 3:11am UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/7 "2013-04-24T03:11:40Z")
> [@Agent\_Foxtrot](#): > > How would this be calculated for a standard 10-round game, since you skip a frame if you hit a strike? Seems to me from some fooling around with this problem that a single frame of n-pin bowling has, for a single bowler, a total number of outcomes equal to the sum, k = 0 to n, of ((n choose k) times (2^k)), or 59,049 for n = 10. That’s only valid for frames 1-9, of course. The 10th frame is a bit trickier, but it’s too late for me to figure this out now. If someone else doesn’t look at this in the next day or so I might, but right now, take the above number to the 9th power and add to it whatever the number of possibilities for the tenth frame is. ---
**Author:** ![Ranger\_Jeff](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ranger_jeff/32/2065_2.png) [@Ranger\_Jeff](https://boards.straightdope.com/u/Ranger_Jeff)\ **Post date:** [April 24, 2013, 4:04am UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/8 "2013-04-24T04:04:53Z")
> [@TriPolar](#): > > !?!? Ok, I get it, different pin combinations. I thought you guys were nuts when the answer wasn’t 10. 1. a gutter ball is a possible outcome also. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 24, 2013, 4:21am UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/9 "2013-04-24T04:21:43Z")
> [@Ranger\_Jeff](#): > > 1. a gutter ball is a possible outcome also. Right. I should have thought they were nuts because the answer was 11. That’ll teach me to try and do maths. ---
**Author:** ![Edward\_The\_Head](https://avatars.discourse-cdn.com/v4/letter/e/ce73a5/32.png) [@Edward\_The\_Head](https://boards.straightdope.com/u/Edward_The_Head)\ **Post date:** [April 24, 2013, 1:27pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/10 "2013-04-24T13:27:12Z")
> [@TriPolar](#): > > In Duck Pins the ball can pass between pins, so I think any combination is possible after the first roll. The ball can not pass between pins in duckpin. At least not if they are all standing, so on the first throw unless you get a gutter ball you’ll knock something down. Well I guess it’s possible if the ball was going slow enough to just bump a pin and not knock it over, but you can’t pass between pins. You can go between a gap on the second and third throw though. I took my daughter ten pin bowling a couple of weeks ago and she left the 7, 8, 9, 10 pins standing. I’ve never seen that before and it looks really strange. ---
**Author:** ![Freddy\_the\_Pig](https://avatars.discourse-cdn.com/v4/letter/f/a587f6/32.png) [@Freddy\_the\_Pig](https://boards.straightdope.com/u/Freddy_the_Pig)\ **Post date:** [April 24, 2013, 3:18pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/11 "2013-04-24T15:18:31Z")
> [@Edward\_The\_Head](#): > > I took my daughter ten pin bowling a couple of weeks ago and she left the 7, 8, 9, 10 pins standing. I’ve never seen that before and it looks really strange. :eek: Did she do that with gutter bumpers up? ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 24, 2013, 3:31pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/12 "2013-04-24T15:31:14Z")
> [@Edward\_The\_Head](#): > > The ball can not pass between pins in duckpin. I’ve seen it happen. Maybe it wasn’t the first roll and a pin was mis-aligned. I only played duckpins as a kid in Maryland but there are still lanes around here. I think there’s a candlepin place around here too. ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [April 24, 2013, 3:44pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/13 "2013-04-24T15:44:35Z")
Of course, with candlepin, you have to worry about where the wood is lying, so you don’t really have a countable discrete set of outcomes. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [April 24, 2013, 3:47pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/14 "2013-04-24T15:47:33Z")
> [@Quercus](#): > > Of course, with candlepin, you have to worry about where the wood is lying, so you don’t really have a countable discrete set of outcomes. Yeah, they don’t clear the pins do they? Never played it, must be hell on the pins. ---
**Author:** ![Edward\_The\_Head](https://avatars.discourse-cdn.com/v4/letter/e/ce73a5/32.png) [@Edward\_The\_Head](https://boards.straightdope.com/u/Edward_The_Head)\ **Post date:** [April 24, 2013, 5:23pm UTC](https://boards.straightdope.com/t/number-of-possible-first-roll-outcomes-in-bowling/656523/15 "2013-04-24T17:23:31Z")
> [@Freddy\_the\_Pig](#): > > :eek: Did she do that with gutter bumpers up? The bumpers were up, but she hit the head pin. She just hit it really slow I guess and the 4,5,6 must have fallen down between between the back ones. It was so strange to see, the people around us were commenting on it as well. They counted it as a split, I wonder if it’s possible to knock all 4 down, I’ll bet it is. > [@TriPolar](#): > > I’ve seen it happen. Maybe it wasn’t the first roll and a pin was mis-aligned. I only played duckpins as a kid in Maryland but there are still lanes around here. I think there’s a candlepin place around here too. Unless the pins were really badly aligned I can’t see it happening on the first roll. On the second or third sure as it happens to me all the damn time. There are still a few duckpin places around in Maryland and I go there more then ten pin. I love duckpin and wish there were more around. Never played candlepin though.