# How to Combine These Two Probabilities?

• I
• Lapse
In summary, the probability that Driver A beats Driver B increases based on the team he is driving for.
Lapse
TL;DR 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.

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

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

Lapse said:
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 ?

Lapse
Lapse said:
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.

Lapse
anuttarasammyak said:
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= .

anuttarasammyak said:
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.

PeroK said:
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.

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.

Lapse said:
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.

Lapse said:
You may.

So you do not have to seek the way to implement the team's effort because you already have it ,i.e.
anuttarasammyak said:
The probability that Team A beats Team B under the same condition of driver= .7576

Lapse
Lapse said:

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.

Lapse
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.

Lapse and PeroK
Dale said:
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

## 1. What is the purpose of combining two probabilities?

Combining two probabilities allows us to calculate the likelihood of two events occurring simultaneously. This can help us make more accurate predictions and decisions based on the combined probability.

## 2. How do I combine two probabilities?

To combine two probabilities, you can use the product rule or the sum rule, depending on the type of events. The product rule is used when the events are independent, while the sum rule is used when the events are mutually exclusive.

## 3. Can probabilities be combined for more than two events?

Yes, probabilities can be combined for any number of events. The same rules apply for combining probabilities of multiple events - product rule for independent events and sum rule for mutually exclusive events.

## 4. What is the difference between independent and mutually exclusive events?

Independent events are those that do not affect each other's probability of occurring. Mutually exclusive events are those that cannot occur at the same time. For example, flipping a coin and rolling a dice are independent events, while getting a head and getting a tail are mutually exclusive events.

## 5. Are there any limitations to combining probabilities?

One limitation to combining probabilities is that it assumes the events are independent or mutually exclusive. In real-life scenarios, this may not always be the case and can affect the accuracy of the combined probability. Additionally, combining probabilities can become more complex when dealing with more than two events.

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