To Black Bird's shock, you and Rainbow Dash kick open the door to your cell having won the game, flown up from the dungeon to the room of the ruins she's in, and you invoking Royal Protocol to arrest Black Bird for marenapping a national celebrity. And flying in the other direction of a designated flying zone. Despite all protests that you started it by mailing a shipment of candy onto her house, Rainbow blows it off - you actually used Black Bird's bank details for the purchase, but nopony else needs to know.
Frustrated at her revenge having not stuck, Black Bird decides it's time to get serious - she pulls out what seems to be a souped-up Mark V Party Mini-Cannon from her mane and decides to just bubblegum you and Rainbow to the floor. She fires and manages to bind you to the wall, but Rainbow is just swift enough to dodge in time, grab three more Party Cannons of her own, and try to fight back.
However, just as the two of them are about to engage in what must surely be the least harmful duel of the century, you hear a knock on the door, and Black Bird goes to answer. At the door is an extremely crotchety old grey stallion, who's just come round to complain about the noise. However, you can see that he's decided to deal with his extra nosy neighbours personally and with extreme prejudice - for he too is carrying a Party Cannon - Mark VII, he must be tech-savvy for an old fogey.
Suddenly Rainbow, Black Bird and the grumpy old neighbour find themselves in a truel. As per prank war tradition (it exists, for Celestia knows what reason) when two or more individuals with Party Cannons gather in one place with disagreements hanging in the air, the matter must be settled with a turn-based shoot-out. Each of you will take turns using a Party Cannon to blast one another, going in a circle - Rainbow, then Black Bird, then the old guy. That means Rainbow is going to have to choose one of the three party cannons she gathered up before, pick one of them, and start playing. Black Bird, with a hint of pride, mutters out loud she hits seven out of ten times. But the old grey pony who ain't what he used to be can hit a pony nine times out of ten.
Black Bird and Old Fogey, as you nickname him, allow Rainbow first shot, as the one with the worst aim. After some sulking over the blunt insult, you just tell Rainbow to get on with it. So, what is she going to do?
Out of the Party Cannons Rainbow grabbed, one is the Mark II which jams about 40% of the time, one's a more modern Mark IV with a jam rate of 20% and then there's the Party Rocket Launcher 2000, which will without fail generate a party for the whole family to not enjoy.
It IS worth noting however that each Party Cannon ever manufactured does come with a certain clause: after one round, if everypony is still gum-free, all the cannons are primed to explode and coat everything in gum. And cake batter. And balloons. As per standard procedure for exact situations like these, you understand.
So what should Rainbow Dash do to ensure that she has the best odds of making it out the duel without getting gummed?
Wizard Ted-Ed duel.
First, we should assume all three knows what each cannon chance of jamming is.
And that both Black and Oldie have perfect logical minds.
Rainbow Dash should use the Mark II and miss on purpose.
(Preferably away from us)
From there it's up to chance.
But she get somewhere from 50% to 66% to win.
(Forgot what it was)
The exact math and logic escapes me at the time, but I know that is the answer.
Well here's the thing
if we gum up old foggy we have a 30% chance to dodge blackbird
If we gum blackbird we get a 10% to dodge fogey
If we miss, than blackbird will go after fogey and its 30% to make it to fogey's turn, at witch its 10% he misses and we all lose
So we have 30% if we hit fogey
10% if we hit blackbird
And 30% × 10% or 3% we all lose
The other 57% of the time we grab the worst cannon and aim at fogey, then pray rainbow misses
the answer to the thumbnail is 16
For the sake of time and space (and sanity) I'll only calculate the times RD wins. Also, BB and OF are acting logically by shooting the person with the greatest accuracy. If there is a 90% and a 30%, you are less likely to get hit by the 30%, so you shoot the 90%.
RD:60%
BB:70%
OF:90%
RD shoots BB, miss:40%, BB shoots OF, miss:30%, OF shoots BB, hit:90%, RD shoots OF, hit:60%, chance:6.48%
(648/10000)
RD shoots BB, miss:40%, BB shoots OF, hit:70%, RD shoots BB, hit:60%, chance:16.8%
(168/1000)
RD shoots BB, hits:60%, OF shoots RD, miss:10%, RD shoots OF, hit:60%, chance:3.6%
(36/1000)
RD vs. BB: wins 26.88%
RD shoots OF, miss:40%, BB shoots OF, miss:30%, OF shoots BB, hit:90%, RD shoots OF, hit:60%, chance:6.48%
(648/10000)
RD shoots OF, miss:40%, BB shoots OF, hit:70%, RD shoots BB, hit:60%, chance:16.8%
(168/1000)
RD shoots OF, hit:60%, BB shoots RD, miss:30%, RD shoots BB, hit:60%, chance:10.8%
(108/1000)
RD vs. OF: wins 34.08%
RD:80%
BB:70%
OF:90%
RD shoots BB, miss:20%, BB shoots OF, hit:70%, RD shoots BB, hit:80%, chance:11.2%
(112/1000)
RD shoots BB, hit:80%, OF shoots RD, miss:10%, RD shoots OF, hit:80%, chance:6.4%
(64/1000)
RD vs. BB: wins 17.6%
RD shoots OF, miss:20%, BB shoots OF, hit:70%, RD shoots BB, hit:80%, chance:11.2%
(112/1000)
RD shoots OF, hit:80%, BB shoots RD, miss:30%, RD shoots BB, hit:80%, chance:19.2%
(192/1000)
RD vs. OF: wins 30.4%
RD:100%
BB:70%
OF:90%
RD shoots BB, hit:100%, OF shoots RD, miss:10%, RD shoots OF, hit:100%, chance:10%
(100/1000)
RD vs. BB: wins 10%
RD shoots OF, hit:100%, BB shoots RD, miss:30%, RD shoots BB, hit:100%, chance:30%
(300/1000)
RD vs. OF: wins 30%
Your best chance is to try to slam Old Fogey with the 60% accuracy.
This all assumes that RD can't waste her first shot.