Question

A fire detection device utilizes three temperature sensitive cells acting independently of each other in such a manner that any one or more may activate the alarm. Each cell possesses a probability of p=.8 of activating the alarm when the temperature reaches 100 degrees celsius or more. let Y equal the number of cells activating the alarm when the temperature reaches 100 degrees.
a) find the probability distribution for Y.
b) find the probability that the alarm will function when the temperature reaches 100 degree.

Answer 1

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:

$P(X=x)=\frac{n!}{(n-x)!\times x!}\times p^x\times (1-p)^{n-x}$

The probability for not activating the alarm is,

$P(Y=0)=\frac{3!}{(3-0)!\times 0!}\times 0.8^0\times (1-0.8)^{3-0}$

$P(Y=0)=0.008$

The probability for activating the alarm by 1 cell is,

$P(Y=1)=\frac{3!}{(3-1)!\times 1!}\times 0.8^1\times (1-0.8)^{3-1}$

$P(Y=1)=0.096$

The probability for activating the alarm by 2 cell is,

$P(Y=2)=\frac{3!}{(3-2)!\times 2!}\times 0.8^2\times (1-0.8)^{3-2}$

$P(Y=2)=0.384$

The probability for activating the alarm by 3 cell is,

$P(Y=3)=\frac{3!}{(3-3)!\times 3!}\times 0.8^3\times (1-0.8)^{3-3}$

$P(Y=3)=0.512$

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.

$P(Y\geq 1)=1-p(Y=0)$

$P(Y\geq 1)=1-0.008$

$P(Y\geq 1)=0.992$

The probability that the alarm will function when 100 degrees is reached is 0.992.