Answer:
P=0.04635.
Step-by-step explanation:
We know that solar-heat installations successfully reduce the utility bill 60% of the time. We calculate the probability that at least 9 out of 10 solar-heat installations are successful and will reduce the utility bill.
We know that: 60%=0.6=p and n=10.
We have a binomial distribution: X : (10, 0.6).
We use the formula to calculate the probability:
[tex]\boxed{ P(X=x)=\left( \begin{array}{c} n \\ x \end{array} \right) \cdot p^x \cdot (1-p)^{n-x}}[/tex]
We get:
[tex]P(X\geq 9)=1-P(X<9)\\\\P(X\geq 9)=1-P(X\leq 8)\\\\P(X\geq 9)=1-\sum_{x=0}^8 P(X=x)\\\\P(X\geq 9)=1-\sum_{x=0}^8 \left( \begin{array}{c} 10 \\ x \end{array} \right) \cdot 0.6^x \cdot (1-0.6)^{10-x}\\\\P(X\geq 9)=1-(0.00011+0.00157+0.01062+0.04247+0.11148+0.20066+0.25082+0.21499+0.12093)\\\\P(X\geq 9)=1-0.95365\\\\P(X\geq 9)=0.04635[/tex]
Therefore, the probability is P=0.04635.
