Notation:
The results of the tests are $T_1,T_2,T_3 \in \{0,1\}$. We indicate a positive test by $T_i = 1$ and zero otherwise, for $i \in \{1,2,3\}$. Similarly, suppose that $Y$ is the status of having covid $(Y=1)$ or not $(Y=0)$.
The accuracy of each test is given by:
- The sensitivity, $\alpha_i = \mathbb{P}(T_i = 1 \mid Y=1)$
- The specificity, $\beta_i = \mathbb{P}(T_i = 0 \mid Y = 0)$.
Problem:
We want to compute the conditional likelihood of infection $(p)$, which is equal to
$$ p = \mathbb{P}(Y=1 \mid T_1 = 1, T_2 = 1, T_3 = 0) $$
For ease of notation let $\mathcal{T}$ denote the event that the tests have two positives and one negative. Then
$$ p = \mathbb{P}(Y = 1 \mid \mathcal{T}). $$
A Bayes' approach:
We can write the probability as:
$$ p = \frac{\mathbb{P}(\mathcal{T} \mid Y = 1)\mathbb{P}(Y = 1)}{\mathbb{P}(\mathcal{T})} $$
The value $\mathbb{P}(Y = 1)$ is the unconditional probability that somebody has COVID, and $\mathbb{P}(\mathcal{T})$ is the probability of observing a set of testing results in the population. The value $\mathbb{P}(\mathcal{T} \mid Y = 1)$ is the likelihood of a observing a test sequence for an infected person.
We need assumptions to figure out each quantity:
- Assumption 1: $\mathbb{P}(Y = 1)$ is equal to the local covid infection rate.
- Assumption 2: The tests $T_1,T_2,T_3$ are independent given $Y$ (the true status)
$$ \mathbb{P}(T_1 = t_1,T_2 = t_2, T_3 = t_3 \mid Y =y) = \prod_{i=1}^3 \mathbb{P}(T_i = t_i \mid Y = y). $$
where $(t_1,t_2,t_3)$ are the values of the test for a specific patient.
Sketch of the computation steps:
First you need to determine the local infection rate (which is not given by your problem). Then you need use the Bayes' formula for $p$.
- You can compute $\mathbb{P}(\mathcal{T} \mid Y = 1)$ by using Assumption 2, and using the appropriate value of sensitivity and specificity for each test. For example, for the third test you need to compute $1-\alpha_3$ to obtain the likelihood of a negative test.
- You can compute $\mathbb{P}(\mathcal{T})$ by the relationship:
$$ \mathbb{P}(\mathcal{T}) = \mathbb{P}(\mathcal{T} \mid Y = 1)\mathbb{P}(Y = 1) + \mathbb{P}(\mathcal{T} \mid Y = 0)\mathbb{P}(Y=0)$$
In Step 2 you have to be careful keeping track of which probabilities you need, and how you can obtain them from $\{\alpha_i,\beta_i)\}_{i=1}^{3}$. Plugging in the values of $(\alpha_i,\beta_i)$ without thinking about the specific sequence of test results in your problem is a potential pitfall.
