Answer :
Answer:
To find the probability that at least one of the 15 randomly chosen homeowners has invested in earthquake insurance, we can use the complement rule. Instead of directly calculating the probability of at least one homeowner having insurance, we'll first calculate the probability that none of the 15 homeowners have insurance, and then subtract this result from 1.
Step-by-step explanation:
Step-by-Step Solution
Probability that a single homeowner has not invested in earthquake insurance:
Given that 3 out of every 10 homeowners have insurance, the probability that a single homeowner does not have insurance is:
(
no insurance
)
=
1
−
(
has insurance
)
=
1
−
3
10
=
7
10
P(no insurance)=1−P(has insurance)=1−
10
3
=
10
7
Probability that none of the 15 homeowners have insurance:
Since the 15 homeowners are chosen randomly and independently, the probability that all 15 homeowners do not have insurance is:
(
none have insurance
)
=
(
7
10
)
15
P(none have insurance)=(
10
7
)
15
Calculate
(
7
10
)
15
(
10
7
)
15
:
Using a calculator,
(
7
10
)
15
≈
0.0047
(
10
7
)
15
≈0.0047
Probability that at least one homeowner has insurance:
The probability that at least one homeowner has insurance is the complement of the probability that none have insurance:
(
at least one has insurance
)
=
1
−
(
none have insurance
)
=
1
−
0.0047
=
0.9953
P(at least one has insurance)=1−P(none have insurance)=1−0.0047=0.9953
Final Answer
The probability that at least one of the 15 homeowners has invested in earthquake insurance is approximately
0.9953
0.9953 or
99.53
%
99.53%.
You can use statistical software or an online tool like the SALT program to verify this calculation and further explore the distribution if needed.
