A vaccine for Covid-19 is known to be $90\%$ effective, i.e. $90\%$ of vaccine recipients are successfully immunised against Covid-19. A new (different) vaccine is tested on $100$ patients and found to successfully immunise $96$ of the $100$ patients. Is the new vaccine better?

Hint: Assume the new vaccine is equally effective as the original vaccine and consider using an appropriate distribution.

I am not sure how to tackle this problem, but my answer is:

Not necessarily, since the sample is $100$ and it is unknown what is the sample of the first vaccine, hence we cannot know whether the second vaccine is better.

What's the correct way to answer this problem?

  • 2
    $\begingroup$ Usually, one is given a confidence level. That is, assume that the second is the same as the first and ask if the probability of the observed outcome is below the specified confidence level. $\endgroup$
    – lulu
    Jul 29, 2021 at 10:38
  • 1
    $\begingroup$ @lulu perhaps "the observed outcome or a more extreme outcome" $\endgroup$
    – Henry
    Jul 29, 2021 at 10:40
  • $\begingroup$ yes, every $ is effective to the producer! :) $\endgroup$
    – Masacroso
    Jul 29, 2021 at 10:51
  • $\begingroup$ It's pertinent to point out this error in the exercise's storyline even though it doesn't affect the thrust of the solution: vaccine effectiveness does not refer to the percentage of recipients successfully immunised!! Nor does it refer to how frequently the vaccine blocks infection, nor the recipient's chance of non-infection (which depends on the disease prevalence, their exposure, safety measures, state of health, etc.). $\endgroup$
    – ryang
    Aug 14, 2021 at 19:53
  • $\begingroup$ Rather, vaccine effectiveness measures increase in protection: $90\%$ effective technically means that in any given situation, one is $10$ times as likely to be infected without the vaccine than with. More details in my explainer. $\endgroup$
    – ryang
    Aug 14, 2021 at 19:53

1 Answer 1


This is an Hypothesis Test exercise.

Consider as the null hypothesis of 90% success a binomial distribution. The extreme probability

$$\mathbb{P}[X\geq 96|p=0.9]\approx 2.40\%$$

Thus you can reject the hypothesis that the old vaccine is better than the new one with a p-value equal or less than 2.4%

this means that the test is significant but not higly significant.

Usually the test is significant if $\text{p-value} < 5%$ and highly significant if $\text{p-value}<1\%$

  • $\begingroup$ Hi, thank you for the answer. How did you calculate 2.40%? $\endgroup$
    – user
    Jul 29, 2021 at 10:59
  • $\begingroup$ @user: $\sum_{x=96}^{100}\binom{100}{x}(0.9)^x(0.1)^{100-x}$ $\endgroup$
    – tommik
    Jul 29, 2021 at 11:16
  • $\begingroup$ thank you. can you answer the other comments? Thank you! $\endgroup$
    – user
    Jul 29, 2021 at 11:19

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