Answer:
To solve this problem, we can use the Poisson distribution formula, given by:
�
(
�
=
�
)
=
�
−
�
�
�
�
!
P(X=k)=
k!
e
−λ
λ
k
where:
�
(
�
=
�
)
P(X=k) is the probability of
�
k events occurring,
�
λ is the average rate of occurrence of the event (in this case, the average number of breakdowns per year),
�
e is the base of the natural logarithm (approximately
2.71828
2.71828),
�
k is the number of events we are interested in (in this case, 5 breakdowns),
�
!
k! is the factorial of
�
k.
Given that the machine breaks down on average 3 times a year, we can use
�
=
3
λ=3 in our calculations.
Now, we need to find the probability of the machine breaking down exactly 5 times in the next 2 years. Since the average rate is given per year, the total rate for 2 years would be
�
total
=
3
×
2
=
6
λ
total
=3×2=6.
Substituting
�
total
=
6
λ
total
=6 and
�
=
5
k=5 into the Poisson distribution formula, we get:
�
(
�
=
5
)
=
�
−
6
×
6
5
5
!
P(X=5)=
5!
e
−6
×6
5
Now, let's calculate this probability:
�
(
�
=
5
)
=
�
−
6
×
6
5
5
!
P(X=5)=
5!
e
−6
×6
5
�
(
�
=
5
)
=
�
−
6
×
7776
120
P(X=5)=
120
e
−6
×7776
Using a calculator or software, we can find:
�
(
�
=
5
)
≈
0.1606
P(X=5)≈0.1606
So, the probability that the machine breaks down exactly 5 times in the next 2 years is approximately
0.1606
0.1606 or
16.06
%
16.06%.
Step-by-step explanation:
