 In this video, we provide the solution to question number 13 for practice exam three for math 1030, in which case, given the weighted voting system with a quota of 11 and five players whose weights are respectively five, four, three, two, one, we need to find the bonds off power distribution for this weighted voting system. So we're gonna start computing all of the winning coalitions possible, right? We have to get up to 11. So let's think of those winning coalitions that involve player one. So if you have player one, for example, you could do player one, player two, they only give you nine, that's not quite enough. You could do player three. So that'll give you player one, player two, and player three. Collectively, they would have 12 points, so that is enough by themselves. But if you lost any one of them, you would not have a winning coalition. So that gives you a critical count of each one, each player is a critical player in that situation. You could also get away with five, four, and two. That adds up to be 11. So you could do player one, player two, and player four. Since the total weight is exactly 11, again, each of the players in that situation are critical. All right, let's see, if you did five, four, and just one, that only gives you 10, so that's not quite good enough. So those are all the minimal ones. I guess you also could do five, three, that gives you eight plus two is 10 plus one. So you could do the coalition, we have player one, player three, player four, player five. Again, in that situation, the exact count is 11. So that would be a winning coalition for which everyone is critical in that situation as well. Those are all the minimal winning coalitions that involve player one. Now, of course, if you add more players to them, that also works. Like if you take this one here, player one, two, and three, you could also do players one, player two, player three, player four. That would be a winning coalition. You could also do players one, two, three, and five. So if you add them, that would be a winning coalition. So just adding people to that one. If this one, if you added player three, you just get back this one, add player five. We haven't done that one yet. Player one, player two, player four, player five. Like so. And then let's see who else. Player one, player three. Again, you could add three, which is said gives you this one, you could add five, which gives you this one. And then, of course, you have everyone. So player one, player two, player three, player four, player five. Like so. And so this gives us, let's see, one, two, three, four, five, six, seven, seven winning coalitions. These are all the winning coalitions. So I was searching first for the minimal winning coalitions. Because after all, if you don't have, if you don't have player five, can you make this work? So you can get four plus three, which is seven, plus two, which is nine, plus one, which is 10. You can't do it. So you do have to have player five. Player five does have veto power. So they're gonna have a huge power distribution going on there, okay? Notice that no other player has veto power in this situation because we can make it work. For example, without player two, and we've seen all the other ones as well. Now we do need to finish our critical counts here. So because player one has veto power, we're gonna see that player one has a critical count in everything. But let's keep on going here. Player two, we're looking at this one. The total count of this thing right here, if you take one, two, three, and four, the total count, let's say you get nine, the plus three is 12, plus two is 14. If you remove two, it's still bigger. If you remove three, it's still bigger. So it turns out that players three and four are not critical on this one, but you do need player two, all right? Because player one, three, and four by themselves don't work. That's not a winning coalition. Then look at the next one here. If you do one, two, three, that is a winning coalition. So you don't need five. If you do one, two, five, that doesn't work on our list. So one, two, five is not a winning coalition. So that makes this one critical. If you do one, three, five, that also doesn't work as well. So you get a critical player there. Looking at this one right here, player one is critical, like I said. If you remove player two, you get player one, four, and five that doesn't win. So you need player two. If you get rid of player four, you get one, two, five, which doesn't work either. And then if you get rid of player five, you do get one, two, four that works. So player five is not critical in that situation. So the next, let's look at the final coalition here. Player one is gonna work. If you remove player two, that does work right here. So player two is not critical. And since player two has four votes, if the other one, if player four didn't work, the other ones aren't gonna work either. So it turns out player one is the only critical one in that one there. So let's then keep track of what we now have done. So with our players, player one, I'll do a different color to make this stand out a little bit better. So player one, player two, player three, player four, and player five. So player one was an E to the seven. So one, two, three, four, five, six, seven. So the critical count for player one is seven. With regard to player two, we get one, two, three, four, and five. So we get five there. For player three, we get one, two, three, and that's just it. So we get three for its critical count. For player four, we get one, two, three as well. And then for player five, we end up with one right here, and that is it. So there's only one like so. So we have to add these all together, seven plus five plus three plus three plus one is 19. So that is the total, that's the total critical count for the whole system there. And so then putting this all together, the bonds off power distribution, your BPI index there, this is going to be seven over 19, which gives us 36.8%, five over 19, which is going to give us 26.3%. You get three over 19, which is going to give us 15.8%. The next one's also three over 19, so that's also 15.8%. And then finally, you get one over 19, which is going to give us 5.3%. And that then gives us the bonds off power distribution for this weighted voting system.