Probability Distribution

Question

There is currently a global pandemic and many researchers and scientists are working towards finding a vaccine that is suitable for children. Assume that Advanced Researcher, a company that is currently testing a few vaccines confirms an 80% chance of effectiveness of one of their vaccines.

a. Which probability distribution would you use to calculate the probability of success or failure of Advanced Researcher's vaccine if it were to be administered to children?

b. What are the assumptions of the probability distribution identified in part "a."?

c. Assume that the vaccine is given to a group of 50 children from Jamaica, what is the probability that it is effective in exactly 40 children? Interpret your results.

Answer 1

a) Since we want to find the probability of success or failure of the vaccine, we can use the Binomial Distribution.

b) The Binomial Distribution has the following assumptions:

1. Each replication of the process results in one of two possible outcomes (success or failure),
2. The probability of success is the same for each replication (In this case 80% or 0.80)
3. There are a fixed number of replications (In this case the 50 children), and
4. The replications are independent, meaning here that a success in one patient does not influence the probability of success in another.

c) The binomial distribution formula is for any random variable X, given by;

P(x:n,p) = ⁿC_x p^x (1-p)^n-x

Here n = 50,

p = 0.8

q = 0.2

x = 40

so P(40:50,0.8) = ⁵⁰C₄₀ 0.8⁴⁰ (0.2)²⁰

P(x = 40) = 0.1398 or 13.98%