To solve this problem, we utilize the properties of the Geometric Distribution, which is appropriate for modeling the number of trials needed to get the first success in a sequence of independent and identically distributed Bernoulli trials.
Let's break down the question step by step:
### a. Defining the Success Event
- Success is defined as the event where a person tests positive for the virus.
### b. Probability of Success ([tex]\( p \)[/tex])
- Given that 2 percent of people test positive for the virus, the probability [tex]\( p \)[/tex] of success (i.e., testing positive) is 0.02.
### c. Defining [tex]\( X \)[/tex]
- [tex]\( X \)[/tex] represents the number of tests needed to get the first positive result.
- We are specifically interested in the probability that the first positive test occurs on the 11th test.
### Applying the Geometric Distribution
We use the formula for the Geometric Distribution, which calculates the probability of having the first success on the [tex]\( k \)[/tex]-th trial:
[tex]\[ P(X = k) = (1-p)^{k-1} \cdot p \][/tex]
By substituting the known values into the formula:
- [tex]\( p = 0.02 \)[/tex]
- [tex]\( k = 11 \)[/tex]
- [tex]\( (1 - p) = 1 - 0.02 = 0.98 \)[/tex]
We calculate the probability as follows:
[tex]\[ P(X = 11) = (0.98)^{10} \cdot 0.02 \][/tex]
This means that we raise 0.98 to the power of 10 (since [tex]\( k-1 = 11-1 = 10 \)[/tex]), and then multiply the result by 0.02.
### Result
The calculated probability of the first positive test occurring on the 11th test is:
[tex]\[ P(X = 11) = 0.016341456137750933 \][/tex]
Thus, the probability that the first positive test will be on the 11th person tested is approximately 0.0163 or 1.63%.
