00:01
For this exercise, we're told that 65 % of a city's population have been infected by a virus.
00:07
So that means for any individual in the city, the probability of infection is 0 .65.
00:13
And we consider a random sample of 12 residents from the city.
00:19
And for part a, we were asked for the probability that half of this sample of 12 have been infected.
00:24
So let's first define a random variable x as the number of residents in the sample of 12 that have been infected.
00:33
Each of the 12 residents can be viewed as a bernoulli trial, two outcomes of interest that you've either been infected or not, and since it's a random sample, their outcomes can be considered to be independent.
00:47
The number of infections in a given number of independent bernoulli trials is a binomial random variable.
00:56
And the probability mass function for the binomial random variable is given by this formula.
01:15
Now for part a, we're looking for the probability that half of the sample has been infected.
01:20
Half of 12 is 6.
01:22
So we're looking for the probability that x is equal to 6.
01:27
Using the probability mass function, this is 12 choose 6 times 0 .65 times 0 .65 to the exponent 6, times 0 .35 to the exponent 6.
01:44
This comes out to 0 .1 to 81 approximately.
01:50
And then for b we are asked for the probability that at least 8 of those selected have been infected.
01:57
This is the probability that x is greater than or equal to 8...