Respuesta :
Using the binomial distribution, we have that:
a) 0.4168 = 41.68% probability that he gets through his career without a serious cycling accident.
b) Considering that the probability of an accident on a given day is independent of any other day, now just the last five years are considered, thus, the probability changes and there is a 0.8825 = 88.25% probability that he will get through his entire career without an accident the same.
For each day, there are only two possible outcomes. Either there is an accident, or there is not. The probability of there being an accident on a day is independent of any other day, which means that the binomial distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, with p probability of a success on each trial.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
- 0.0001 probability of an accident on a given day, thus [tex]p = 0.0001[/tex]
- 250 days, for 35 years, thus the number of days is [tex]n = 250(35) = 8750[/tex].
Item a:
The probability is P(X = 0), thus:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8750,0}.(0.0001)^{0}.(0.9999)^{8750} = 0.4168[/tex]
0.4168 = 41.68% probability that he gets through his career without a serious cycling accident.
Item b:
- The probability changes, as we consider now just five years.
- The number of days is now [tex]n = 250(5) = 1250[/tex].
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{8750,0}.(0.0001)^{0}.(0.9999)^{1250} = 0.8825[/tex]
Considering that the probability of an accident on a given day is independent of any other day, now just the last five years are considered, thus, the probability changes and there is a 0.8825 = 88.25% probability that he will get through his entire career without an accident the same.
A similar problem is given at https://brainly.com/question/24756209
