Answer:
To solve this problem, we can use the binomial probability formula:
�
(
�
=
�
)
=
(
�
�
)
⋅
�
�
⋅
(
1
−
�
)
�
−
�
P(X=k)=(
k
n
)⋅p
k
⋅(1−p)
n−k
Where:
�
(
�
=
�
)
P(X=k) is the probability of getting exactly
�
k successes.
�
n is the number of trials or employed women selected.
�
k is the number of successful trials (in this case, women who have never been married).
�
p is the probability of success in a single trial (the probability that an employed woman has never been married).
(
�
�
)
(
k
n
) is the binomial coefficient, which can be calculated as
�
!
�
!
(
�
−
�
)
!
k!(n−k)!
n!
.
Given:
�
=
0.30
p=0.30 (probability that an employed woman has never been married)
�
=
15
n=15 (number of employed women selected)
Let's calculate the probabilities for the two parts of the question:
A) Probability that none of them have ever been married (
�
=
0
k=0):
�
(
�
=
0
)
=
(
15
0
)
⋅
(
0.30
)
0
⋅
(
1
−
0.30
)
15
=
1
⋅
1
⋅
0.987654321
=
0.9877
P(X=0)=(
0
15
)⋅(0.30)
0
⋅(1−0.30)
15
=1⋅1⋅0.987654321=0.9877
Rounded to 4 decimal places, the probability that none of them have ever been married is approximately 0.9877.
B) Probability that at least one of them has never been married (which is the complement of none of them having never been married):
�
(
At least one
)
=
1
−
�
(
None
)
=
1
−
0.9877
=
0.0123
P(At least one)=1−P(None)=1−0.9877=0.0123
Rounded to 4 decimal places, the probability that at least one of them has never been married is approximately 0.0123.
Explanation: