Respuesta :
Answer:
The required probability distribution is:
Y 0 1 2 3
P(Y) 0.008 0.096 0.384 0.512
The probability that the alarm will function when 100 degrees is reached is 0.992.
Step-by-step explanation:
Consider the provided information.
Part (A)
Y represents the number of cells activating the alarm.
The number of alarms are n=3
Each cell possesses a probability of p=.8 of activating the alarm when the temperature reaches 100 degrees Celsius or more.
The binomial distribution can be calculated as:
[tex]P(X=x)=\frac{n!}{(n-x)!\times x!}\times p^x\times (1-p)^{n-x}[/tex]
The probability for not activating the alarm is,
[tex]P(Y=0)=\frac{3!}{(3-0)!\times 0!}\times 0.8^0\times (1-0.8)^{3-0}[/tex]
[tex]P(Y=0)=0.008[/tex]
The probability for activating the alarm by 1 cell is,
[tex]P(Y=1)=\frac{3!}{(3-1)!\times 1!}\times 0.8^1\times (1-0.8)^{3-1}[/tex]
[tex]P(Y=1)=0.096[/tex]
The probability for activating the alarm by 2 cell is,
[tex]P(Y=2)=\frac{3!}{(3-2)!\times 2!}\times 0.8^2\times (1-0.8)^{3-2}[/tex]
[tex]P(Y=2)=0.384[/tex]
The probability for activating the alarm by 3 cell is,
[tex]P(Y=3)=\frac{3!}{(3-3)!\times 3!}\times 0.8^3\times (1-0.8)^{3-3}[/tex]
[tex]P(Y=3)=0.512[/tex]
Hence, the required probability distribution is:
Y 0 1 2 3
P(Y) 0.008 0.096 0.384 0.512
Part (b) Find the probability that the alarm will function when the temperature reaches 100 degree.
[tex]P(Y\geq 1)=1-p(Y=0)[/tex]
[tex]P(Y\geq 1)=1-0.008[/tex]
[tex]P(Y\geq 1)=0.992[/tex]
The probability that the alarm will function when 100 degrees is reached is 0.992.
By using binomial distribution formula we got that probability distribution for Y is as following :
Y 0 1 2 3
P(Y) 0.008 0.096 0.384 0.512
and the probability that the alarm will function when the temperature reaches 100 degree is 0.992
What is probability ?
Probability is chances of occurring of an event.
Here given that Each cell possesses a probability of p=.8 of activating the alarm when the temperature reaches 100 degrees celsius or more.
we know that binomial distribution can be calculated as:
[tex]P(X=x)=\frac{n !}{(n-x) ! x !} p^{x} (1-p)^{n-x}[/tex]
Where
p = probability of success = 0.8
q= probability of failure =1-p=1 - 0.8 = 0.2
n is total number= 3
Let x=0 denote for not activating the alarm , x=1 denote activating the alarm by 1 cell , x=2 denote activating the alarm by 2 cell and x=3 denote activating the alarm by 3 cell
Hence
[tex]$\begin{aligned}&P(Y=0)=\frac{3 !}{(3-0) ! \times 0 !} \times 0.8^{0} \times(1-0.8)^{3-0} \\&P(Y=0)=0.008\end{aligned}$[/tex]
[tex]$\begin{aligned}&P(Y=1)=\frac{3 !}{(3-1) ! \times 1 !} \times 0.8^{1} \times(1-0.8)^{3-1} \\&P(Y=1)=0.096\end{aligned}$[/tex]
[tex]$\begin{aligned}&P(Y=2)=\frac{3 !}{(3-2) ! \times 2 !} \times 0.8^{2} \times(1-0.8)^{3-2} \\&P(Y=2)=0.384\end{aligned}$[/tex]
[tex]$\begin{aligned}&P(Y=3)=\frac{3 !}{(3-3) ! \times 3 !} \times 0.8^{3} \times(1-0.8)^{3-3} \\&P(Y=3)=0.512\end{aligned}$[/tex]
Hence probability distribution can be written as :
Y 0 1 2 3
P(Y) 0.008 0.096 0.384 0.512
Now the probability that the alarm will function when the temperature reaches 100 degree can be calculated as
[tex]\begin{aligned}&P(Y \geq 1)=1-p(Y=0) \\&P(Y \geq 1)=1-0.008 \\&P(Y \geq 1)=0.992\end{aligned}[/tex]
By using binomial distribution formula we got that probability distribution for Y is as following :
Y 0 1 2 3
P(Y) 0.008 0.096 0.384 0.512
and the probability that the alarm will function when the temperature reaches 100 degree is 0.992
To learn more about probability visit : brainly.com/question/24756209
