The cricket toss gives each team a 50% chance of being able to choose whether they bat or bowl first. I think this is an illusion of control, and there's no advantage to winning it. I'll try and explain why.
Let's say we change the toss so that it's a simple "Heads = home team bats first, Tails = away team bats first". Let's say the home team wants to bat first. Then they have a 50% chance of getting what they want. This is equivalent to them winning the toss and electing to bat.
These two situations result in the same probability that a given team will start with what they want - so why do we use the (slightly) more convoluted method of giving the toss winner the option to choose? If we did a simple "H = home team bats, T = away team bats", then the overall outcome would be the same, and it would remove all this debate around what kind of advantage winning the toss gives (because there is no "winning" the toss, you either get what you want or you don't).
Is there something wrong with my analysis?