Vaccine safety

In a period of 14 days the probability for any random individual to get condition X is $\frac{1}{N}$ so the probability to get exactly k cases is $P_{(c=k)}=(1-\frac{1}{N})^{N-k}\cdot(\frac{1}{N})^k\cdot {N \choose k}$

Since $\lim_{N \rightarrow \infty} (1-\frac{1}{N})^N=1/e$ , for very large $N$ and $k << N$ , ${N \choose k}=\frac{N!}{(N-k)!k!}\approx \frac{N^{k}}{k!}$ , this can be well approached by $P_{(c=k)}\approx \frac{1}{ek!}$ So that when $N \rightarrow \infty$ : $P_{(c \le 6)} = \frac{1}{e}(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\frac{1}{6!})=\frac{1957}{720 \cdot e}=0.999916758...$ Hence $P_{(c \ge 7)} = 1- P_{(c \le 6)} = 0.000083241....$ And $\lfloor 10^8 P_{(c \ge 7)}\rfloor =8324$

Some values for finite $N$ : For $N=1000, P_{(c \ge 7)}=8.197×10^{-5}$ , for $N=10000, P_{(c \ge 7)}=8.311×10^{-5}$ , for $N=100000, P_{(c \ge 7)}=8.323×10^{-5}$ and for $N=1000000, P_{(c \ge 7)}=8.324×10^{-5}$ .

The probability is very, very low that the null hypothesis is true, and it can be rejected. This implies that there is an effect of the vaccine on the possibility of developing condition X. This does not mean that the vaccine is not safe. For the average person, the risk of developing condition X after vaccination is about the same as the risk to develop it spontaneously during any period of about 100 days. If the vaccine protects against a real health risk, the benefit is much, much more important than the drawback.