Undergrad How to Combine These Two Probabilities?

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TL;DR

Combination of two probabilities such that outcome of 2nd probability increases outcome of the 1st probability.

I am trying to determine the likelihood of a driver winning a race based on an associated rating as well as the team he drives for.

The probability that Driver A beats Driver B = .8504
The probability that Team A beats Team B = .7576

How do I combine these two probabilities, where the outcome is an increase in Driver A's probability of winning?
The point is that Driver A is a better driver than Driver B, but the team he is racing for produces a better car. Thus, Driver A's probability of winning increases. If Driver A and Driver B switched cars, then Driver A's probability of winning decreases.

It may be the case that the proper way to implement the team's effect is to actually insert its probability into the equation between the drivers. In that case, I am using a standard Elo calculation.

Example:
Driver A Rating = 1543
Driver B Rating = 1241

(Driver B - Driver A) | (1241 - 1543) = -302
(Result / 400) | (-302 / 400) = -.755
(1 + 10^Result) | (1 + 10^-.755) = 0.175792
(1 / (1 + Result)) | (1 / (1 + 0.175792)) = 0.8504, or an 85% chance of beating Driver B.

 

[anuttarasammyak]

Summary:: Combination of two probabilities such that outcome of 2nd probability increases outcome of the 1st probability.

The probability that Driver A beats Driver B = .8504
The probability that Team A beats Team B = .7576

May I interpret them as
The probability that Driver A beats Driver B under the same condition of car = .8504
The probability that Team A beats Team B under the same condition of driver= .7576 ?

 

[PeroK]

Summary:: Combination of two probabilities such that outcome of 2nd probability increases outcome of the 1st probability.

I am trying to determine the likelihood of a driver winning a race based on an associated rating as well as the team he drives for.

The probability that Driver A beats Driver B = .8504
The probability that Team A beats Team B = .7576

How do I combine these two probabilities, where the outcome is an increase in Driver A's probability of winning?
The point is that Driver A is a better driver than Driver B, but the team he is racing for produces a better car. Thus, Driver A's probability of winning increases. If Driver A and Driver B switched cars, then Driver A's probability of winning decreases.

It may be the case that the proper way to implement the team's effect is to actually insert its probability into the equation between the drivers. In that case, I am using a standard Elo calculation.

There's no "proper" way to combine these probabilities because you need to decide the relative importance of team/car and driver. Presumably $0.7576$ represents the probability that Car A beats Car B when they are both driven by drivers of equal ability. And, $0.8504$ represents the probability that driver A beats driver B when they both drive identical cars.

This does not (and cannot logically) completely define the probabilities when you start mixing cars and drivers of different abilities. You have a choice of additional hypotheses.

 

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May I interpret them as
The probability that Driver A beats Driver B under the same condition of car = .8504
The probability that Team A beats Team B under the same condition of driver= .

May I interpret them as
The probability that Driver A beats Driver B under the same condition of car = .8504
The probability that Team A beats Team B under the same condition of driver= .7576 ?

You may.

 

[Lapse]

There's no "proper" way to combine these probabilities because you need to decide the relative importance of team/car and driver. Presumably $0.7576$ represents the probability that Car A beats Car B when they are both driven by drivers of equal ability. And, $0.8504$ represents the probability that driver A beats driver B when they both drive identical cars.

This does not (and cannot logically) completely define the probabilities when you start mixing cars and drivers of different abilities. You have a choice of additional hypotheses.

I see your point.

What then would be the most logical way to derive probabilities for this scenario?
If we have two drivers with no previous experience, and two cars with no previous experience, then we might say that all four factors have a starting Elo of 1000.

If Driver A with Team A then beats Driver B with team B, how are their ratings adjusted such that we can determine their probabilities for the next race?

How I was attempting to derive this before was by using the probability of a team beating another team in order to determine the driver's probability of winning.

Here is a practical example. Let Driver A, whom is the best driver, be placed in Team Z, the worst team. Then, let's take Driver Z, one of the poorer drivers, and stick him in Team A, the best team. We can actually assume that Driver Z will win, because the driver's themselves are so close in skill level relative to their cars' differences.

If the above example is the case, then the team will play a majority role in determining the driver probabilities.

 

[anuttarasammyak]

It may be the case that the proper way to implement the team's effect is to actually insert its probability into the equation between the drivers. In that case, I am using a standard Elo calculation.

You may.

So you do not have to seek the way to implement the team's effort because you already have it ,i.e.

The probability that Team A beats Team B under the same condition of driver= .7576

 

[PeroK]

I see your point.

What then would be the most logical way to derive probabilities for this scenario?

I know nothing about motor racing!

You need a model of some sort.

 

[Dale]

You need to have some sort of model that would allow you to break the probability into different components. Usually this would be some sort of a latent trait, like the Elo model. You could assume that Elo scores for driver and team are additive. Or you could transform to a log odds scale and assume that the log odds are additive.

In any case, you will have to assume some model, and with only two points of data pretty much any model you can think of will be compatible with the data. So with such limited data you cannot expect that your model will be realistic.

 

[Dale]

Or you could transform to a log odds scale and assume that the log odds are additive.

FYI, doing this is probably the easiest approach and it gives a combined probability of 0.947