Question about Bayesian probabilities. Where did I go wrong? I was solving a question for an acquaintance, about probabilties. The original question goes like this:

Of all threats in an year, $12\%$ are Tier 1 and the remaining $88\%$ are Tier 2. If the probability that a reported Tier 1 threat is actually a Tier 2 is $21\%$ and the probability that a reported Tier 2 threat is actually a Tier 1 is $33\%$, then what is the probability that a reported Tier 1 is actually a Tier 1?


Here are my attempts to solve;

Let probability that an event is Tier 1$=P(T_1)=12\%$
Let probability that an event is Tier 2$=P(T_2)=88\%$
Let probability that a reported Tier 1 was actually Tier 2$=P(M_{1\to2})=21\%$
Let probability that a reported Tier 2 was actually Tier 1$=P(M_{2\to1})=33\%$
Let there be $10000$ events.

...

Then,
$\#T_1=1200$
$\#T_2=8800$

...

Let the number of reported Tier 1 events be $O$.
Let the number of reported Tier 2 events be $T$.

...

Then,
Number of actual Tier 1 Events from number of reported Tier 1 events $={{(100-21)}\over100}\times O=O_1$
Number of actual Tier 2 Events from number of reported Tier 1 events $={{21}\over100}\times O=T_1$
Number of actual Tier 1 Events from number of reported Tier 2 events $={{33}\over100}\times T=O_2$
Number of actual Tier 2 Events from number of reported Tier 2 events $={{(100-33)}\over100}\times T=T_2$

...

Since,
Number of Tier 1 threats $=1200$
$O_1+O_2=1200$
So $\left({{(100-21)}\over100}\times O\right)+\left({{33}\over100}\times T\right)=1200$
Since,
Number of Tier 2 threats $=8800$
$T_1+T_2=8800$
So $\left({{21}\over100}\times O\right)+\left({{(100-33)}\over100}\times T\right)=8800$

From here I get two equations in $O$ and $T$,
$\left({{79}\over100}\times O\right)+\left({{33}\over100}\times T\right)=1200$
and
$\left({{21}\over100}\times O\right)+\left({{67}\over100}\times T\right)=8800$
Wolfram|Alpha reports that $O$ is negative.
How is this possible? Where did I go wrong?

The actual question is near the middle here.
The answer should be explainable in a text-only environment.

Using conditional probability expressions in a Bayesian probability question will lead to much less confusion about the topic.
Use $T_1,T_2$ as the mutually exclusive and exhaustive events that a threat is tier 1 or tier 2 respectively, and $R_1,R_2$ as the m.e.e. events that a threat is reported as such.

Of all threats in an year, $12\%$ are Tier 1 and the remaining $88\%$ are Tier 2. If the probability that a reported Tier 1 threat is actually a Tier 2 is $21\%$ and the probability that a reported Tier 2 threat is actually a Tier 1 is $33\%$, then what is the probability that a reported Tier 1 is actually a Tier 1?

So, you are told :


*

*$\mathsf P(T_1)=0.12$ the probability that an event is actually tier 1

*$\mathsf P(T_2)=0.88$ the probability that an event is actually tier 2 

*$\mathsf P(T_2\mid R_1)=0.21$ the probability that an event is tier 2 given that it is reported as tier 1.

*$\mathsf P(T_1\mid R_2)=0.33$ the probability that an event is tier 1 given that it is reported as tier 2. 


You seek $\mathsf P(T_1\mid R_1)$, the probability that a threat is tier 1 given that it is reported as tier 1.

 $\mathsf P(T_1\mid R_1)=1-\mathsf P(T_2\mid R_1)$.

