Respuesta :
Answer:
(a) P = 0.0167
(b) P = 0.9833
(c) P = 0.0093
Explanation:
To answer these questions, we will use the binomial distribution because we have n identical events (13 households) with a probability p of success (27% still have a traditional phone landline). So, the probability that x families has a traditional phone landline can be calculated as
[tex]\begin{gathered} P(x)=nCx\cdot p^x\cdot(1-p)^x \\ \\ \text{ Where nCx = }\frac{n!}{x!(n-x)!} \end{gathered}[/tex]Replacing n = 13 and p = 27% = 0.27, we get:
[tex]P(x)=13Cx\cdot0.27^x\cdot(1-0.27)^x[/tex]Part (a)
Then, the probability that no families have a phone landline can be calculated by replacing x = 0, so
[tex]P(0)=13C0\cdot0.27^0\cdot(1-0.27)^{13-0}=0.0167[/tex]Part (b)
The probability that at least one family has a phone landline can be calculated as
[tex]\begin{gathered} P(x\ge1)=1-P(0) \\ P(x\ge1)=1-0.167 \\ P(x\ge1)=0.9833 \end{gathered}[/tex]Part (c)
The probability that at least eight families have a phone landline can be calculated as
[tex]P(x\ge8)=P(8)+P(9)+P(10)+P(11)+P(12)+P(13)[/tex]So, each probability is equal to
[tex]\begin{gathered} P(8)=13C8\cdot0.27^8\cdot(1-0.27)^{13-8}=0.0075 \\ P(9)=13C9\cdot0.27^9\cdot(1-0.27)^{13-9}=0.0015 \\ P(10)=13C10\cdot0.27^{10}\cdot(1-0.27)^{13-10}=0.0002 \\ P(11)=13C11\cdot0.27^{11}\cdot(1-0.27)^{13-11}=0.00002 \\ P(12)=13C12\cdot0.27^{12}\cdot(1-0.27)^{13-12}=0.000001 \\ P(13)=13C13\cdot0.27^{13}\cdot(1-0.27)^{13-13}=0.00000004 \end{gathered}[/tex]Then, the probability is equal to
P(x≥8) = 0.0093
Therefore, the answers are
(a) P = 0.0167
(b) P = 0.9833
(c) P = 0.0093
