Firstly, I'd like to apologize because my comments definitely provided more criticism than constructive commentary. (Although hopefully a healthy blend of both).

Secondly, I think the general answer to your question is: Yes, Bayesian inference can be used in this manner where you take known quantities $P($Goalie is Sick$)$, $P($Goal Keeper Is Sick | USA Wins World Cup$)$, $P($USA Wins World Cup$)$ to make an updated statement of the likelihood of a win with the updated information that the USA goalie is in fact sick.

The issue here is not the mathematics behind the relationship of these events, as much as it is your ability to accurately estimate the assumptions you are using originally. Events such as $P($USA Wins World Cup$)$ are not easily quantifiable, and may require an extreme magnitude of assumptions to even arrive at a number (all teams are equal, all goalies have same likelihood of sickness, etc).

So, under the assumption that you truly know the probability of the axiomatic events, then yes, you may use Bayesian inference in this way.

Typically, Bayesian inference is used not for predictions in the manner, but for discrediting extreme observations for scenarios with small data sets. Where you make an assumption that the random process you are observing comes from a distribution $P(X|\theta)$ and a further assumption that your $\theta$ parameter is in fact a random variable. I.E.$\exists \theta_1, \theta_2$ s.t. $P(\theta_1) \neq P(\theta_2)$ and $P(\theta_1),P(\theta_2)>0$.

For example, consider the case of trying to estimate the height of the next individual that you encounter. A naive approach would be to average the heights of all individuals that you have encountered and use this as your estimate. If you had more information, say the gender of the individual, you could make much more accurate estimates due to the fact that the heights of humans varies quite differently (in both mean and variance) for the two genders.

Firstly, I'd like to apologize because my comments definitely provided more criticism than constructive commentary. (Although hopefully a healthy blend of both).

Secondly, I think the general answer to your question is: Yes, Bayesian inference can be used in this manner where you take known quantities $P($Goalie is Sick$)$, $P($Goal Keeper Is Sick | USA Wins World Cup$)$, $P($USA Wins World Cup$)$ to make an updated statement of the likelihood of a win with the updated information that the USA goalie is in fact sick.

The issue here is not the mathematics behind the relationship of these events, as much as it is your ability to accurately estimate the assumptions you are using originally. Events such as $P($USA Wins World Cup$)$ are not easily quantifiable, and may require an extreme magnitude of assumptions to even arrive at a number (all teams are equal, all goalies have same likelihood of sickness, etc).

So, under the assumption that you truly know the probability of the axiomatic events, then yes, you may use Bayesian inference in this way.

Typically, Bayesian inference is used not for predictions in the manner, but for discrediting extreme observations for scenarios with small data sets. Where you make an assumption that the random process you are observing comes from a distribution $P(X|\theta)$ and a further assumption that your $\theta$ parameter is in fact a random variable. I.E.$\exists \theta_1, \theta_2$ s.t. $P(\theta_1),P(\theta_2)>0$.

For example, consider the case of trying to estimate the height of the next individual that you encounter. A naive approach would be to average the heights of all individuals that you have encountered and use this as your estimate. If you had more information, say the gender of the individual, you could make much more accurate estimates due to the fact that the heights of humans varies quite differently (in both mean and variance) for the two genders.