The Question
A particular Covid-19 vaccine produces a side effect (fever) with probability $p \in[0,1]$. A clinic vaccinates 500 people each day and records the number of people having side effects on 5 randomly chosen days.
(i) What is the probability that the number of people who developed side effects on the 5 observed days are $10,15,0,20,25 ?$
(ii) Find the value of $p$ that maximizes the probability obtained in part (i).
(iii) Based on the value of $p$ obtained in part (ii), what is the expected number of people who will develop side effects on each day?
My Attempt
1.) $P(\text{10 side effects on Day 1}) = {500 \choose 10}p^{10}(1-p)^{490}$
Since, 10 people are already recorded to have side effects, on day two the number of people being checked is 490.
$P(\text{15 side effects on Day 2}) = {490 \choose 15}p^{15}(1-p)^{475}$
$P(\text{0 side effects on Day 3}) = p^{0}(1-p)^{475}$
$P(\text{20 side effects on Day 4}) = {475 \choose 20}p^{20}(1-p)^{455}$
$P(\text{25 side effects on Day 5}) = {455 \choose 25 }p^{25}(1-p)^{430}$
So the total Probability that the number of people who developed side effects on the 5 observed days are $10,15,0,20,25 = K \cdot p^{70}(1-p)^{2325}$ where $K$ is some positive integer.
2.)
After differentiating, the above expression is $0$ when $p = \frac{70}{2395}$. So, the above expression is maximized at $p = \frac{70}{2395}$
3.) I am confused about the expectation part, please help. Do I have to fix a particular day and calculate expectation on that day? But won't that depend on how many patients have been recorded earlier? Kindly help with the third part.
