This feels like a stupid question but I’m going to ask it anyways. I’m using fantasy football as an example but this isn’t a Game Room question as it could apply to any number of situations.
I’m playing my daughter in Fantasy Football and according to the app, she currently has a 55% chance of winning and I have a 45% chance. Does she have a 10% greater chance of winning or 5%? Obviously 55 is 10 more than 45 but I only need to increase my chances by 5% in order for the probability to be 50:50. So is it 5, 10, or something else?
If she had 55 jelly beans and you had 45 jelly beans, I would say that she had 10 more jelly beans than you, even though she would only have to give you 5 of them for the two of you to be equal.
Something else.
The point above is that “percentage” isn’t a unit. It is a way of expressing a number. 55% is semantically identical to the number 0.55. Change the question to use 0.55 and 0.45 and one source of possible confusion is eliminated.
If it were just a matter of what fraction of 0.45 needs adding to make 0.55 the answer would be {{0.55 - 0.45}\over{0.45}} = 2/9 = 0.222… or about 22.2%.
But as you note, in order to become equal, since the opposition’s chance diminishes by the same amount , it is half that, so about 11.1%
By my reckoning, as a gambler, your daughter is 22.2% (55/45 - 1) more likely than you to win. She is expected to win 55 of every 100 contests between you.
So, to have a bet I would need better than $1.82 (-122) about your daughter, or better than $2.22 (+122) about your chances to back you.
Her chance of winning is 10 percentage points higher than yours. As the previous two posters have calculated that means she has a 22.2% higher chance of winning than you. That though is the percentage of a completely different thing what your 45% chance of winning is a percentage of.
Vise versa you have a 18.2% less chance of winning that she has. % more and % less are comparisons fraught with potential confusion. % more and % less of a percentage doubly so.
I suspected that the actual answer was something different.
To expand on @Thudlow_Boink’s jelly bean scenario. Suppose that I’m playing a game where each player starts with 50 jelly beans. A coin is tossed, I call heads and it is heads. My opponent gives me a jelly bean. The score is now 51-49. Would you say that I’m ahead by 1 or 2?
This reminds me of the situation where two teams are in direct competition to win their division and make the playoffs. If one of those teams beats the other, it sort of helps them twice, because it’s a win for them and a loss for their rival.
If one party is ahead by 51-49, they are ahead by two votes. It only takes one vote to switch for there to be a tie.If they were ahead 55-45 then it would require five votes to switch to create a tie.
Don’t let semantics get in the way. Who is ahead and by how much is a different question that how many switches would it take to create a tie. Different questions can have different answers.
On the other hand, if the party that’s up by two votes has a senator suddenly die or retire or change to a third party, they’re not tied; they’re still up by 1.
Likewise, your daughter’s chance of winning could decrease without yours increasing, if another player joined the game (which may or may not be possible, depending on the game).
As @Chronos notes, whether it’s a useful analogy depends on the nature of the game: whether it’s possible for one party to lose a “point” without the other simultaneously gaining one.