# Calculating Vaccine Effectiveness with Bayes’ Theorem

We can use Bayes’ Theorem to estimate the probability of someone not having an effect (meaning they get infected after vaccination) for both Covishield and Covaxin, considering a population of 1.4 billion individuals.

Assumptions:

- We assume equal distribution of both vaccines in the population (700 million each).
- We focus on individual protection probabilities, not overall disease prevalence.

Calculations:

Covishield:

- Prior Probability (P(Effect)): Assume 10% of the vaccinated population gets infected (no effect), making P(Effect) = 0.1.
- Likelihood (P(No Effect|Effect)): This represents the probability of someone not being infected given they received Covishield. Given its 90% effectiveness, P(No Effect|Effect) = 0.9.
- Marginal Probability (P(No Effect)): This needs calculation, considering both vaccinated and unvaccinated scenarios. P(No Effect) = P(No Effect|Vaccinated) P(Vaccinated) + P(No Effect|Unvaccinated) * P(Unvaccinated) Assuming 50% effectiveness for unvaccinated individuals and equal vaccination rates, P(No Effect) = (0.9 0.5) + (0.5 0.5) = 0.7.

Now, applying Bayes’ Theorem:

P(Effect|No Effect) = (P(No Effect|Effect) * P(Effect)) / P(No Effect) P(Effect|No Effect) = (0.9 0.1) / 0.7 ≈ 0.129

Therefore, about 12.9% of people vaccinated with Covishield could still get infected, meaning 700 million * 0.129 ≈ 90.3 million individuals might not have the desired effect from the vaccine.

Covaxin:

Similar calculations for Covaxin, with its 78–81% effectiveness range, would yield a range of 19.5% — 22.2% for the “no effect” probability. This translates to potentially 136.5 million — 155.4 million individuals not fully protected by Covaxin in the given population.

Important Note:

These are hypothetical calculations based on limited assumptions. Real-world effectiveness can vary depending on individual factors, virus strains, and vaccination coverage.

Conclusion:

Both Covishield and Covaxin offer significant protection against COVID-19, but they are not 100% effective. A significant portion of the vaccinated population might still have some risk of infection. Vaccination remains crucial for reducing disease spread and severe outcomes, but additional precautions like hand hygiene and masks might be advisable.