Suppose a person infected with coronavirus went to a maskless indoor party with 11 other people. Of the 11 people, 4 stood 3 feet apart from them and talked loudly with them for an hour. The other 7 people maintained a distance of at least six feet throughout the entire party. Suppose the infected person has a 40% chance of infecting people who stand within 3 feet and a 10% chance of infecting people who are in the same room but maintain a distance of at least 6 feet. Let I be the number of people the person infects at the party. Find E[I] 

Solution:

Let X be the number of people who became infected at the party and were within 3 feet of the infected.
As the probability of success is constant, we have that X follows a Binomial distribution with the following parameters:

p(success)=0.4

n=4

The mean or expected value for a Binomial distribution is defined as follows:

μ=E(x)=n.p

E(X)=4(0.4)=1.6

Let Y be the number of people who were infected at the party and were at least 6 feet from the infected.
As the probability of success is constant, we have that Y follows a Binomial distribution with the following parameters:

p(success)=0.1

n=7

Then:

μ=E(Y)=n.p

E(Y)=7(0.1)=0.7

Let I be the number of people that the person infects at the party, we then have that to find the expected value of I we must add the expected value of X with the expected value of Y, thus we have:

E(I)=E(X)+E(Y)

E(I)=1.6+0.7=2.3

The person is expected to infect 2.3 people at the party.

Step-by-step explanation

Expected value for a Binomial variable:

μ=E(X)=n.p