After noting that the average age for Americans receiving the MMR vaccine is 1.2 years old, let us use an
exponential model to allow for unvaccinated children. What is the probability that a 5-year-old is
unvaccinated?"

2) P(5-year-old child is unvaccinated) = 0.01551

Step-by-step explanation

In Probability, exponential distributions are often used to model the time between events. In our question, the "event" here is when a child gets vaccinated.

The exponential distributions has density function given by:  fX?(x)=λe−λx   for x>0  (and 0 for x≤0 )

The mean of the exponential distribution is given by λ1?. According to our problem, the average time is 1.2 (in years). Solving for lambda, we have:

λ1?=1.2

1=1.2λ

1.21?=65?=λ

λ=0.8333

Lastly, we compute for the probability that no event (vaccination) happened to a child of 5 years. This is the same as computing for the probability that x is greater than 5.

P(x>5)=1−P(X≤5)

                    =1−∫05?fX?(x)dx

                    =1−∫05?λe−λxdx

                    =1−∫05?65?e−65?xdx

                     =1−0.98449

                     =0.01551