How to find the probability of a podium finish

Say in a race there are 22 competitors and say that the probability of driver A of winning the race is 55%. How can one calculate the probability of driver A to finish the race in the first three positions?

With just what you've provided, the only answer is 55% for driver A to finish in the first three positions (actually, exactly first place). This is because we do not know driver A's chances of ending up in 2nd place, 3rd place, or not in the top three positions.

However, I feel what you mean leans more towards what is the probability of driver A finishing in 1st, 2nd, OR 3rd place, right?
I don't think there is enough information here. A wins 55%, so the probability of coming in first, second or third is clearly greater than .55.

A does not win 45% of the time. We could estimate A will come in second place 55% of the time when it does not win. That is, there is approximately 55% chance A will beat the other 20 horses.

.55 * .45 = .2475

So A comes in first place .55 and second place .2475

So A will not come in first or second (1 - (.55 + .2475)) = .2025

When A does not come in first or second, it will come in third place approximately 55%

So A will come in first or second or third approximately

.55 + (.55*.45) + (.55*.2025) = .9089

or 90.89% of the time