Answer:
(a) The value of P (None) is 0.062.
(b) The value of P(at least one) is 0.938.
(c) The value of P(at most one) is 0.253.
(d) The event is not unusual.
Step-by-step explanation:
Let <em>X</em> = number of households watching the show.
The probability of the random variable <em>x</em> is, P (X) = <em>p</em> = 0.18.
The sample selected for the survey is of size, <em>n</em> = 14
The random variable <em>X</em> follows a Binomial distribution with parameter <em>n</em> = 14 and <em>p</em> = 0.18.
The probability of a Binomial distribution is computed using the formula:
![P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0,1,2,...](https://tex.z-dn.net/?f=P%28X%3Dx%29%3D%7Bn%5Cchoose%20x%7Dp%5E%7Bx%7D%281-p%29%5E%7Bn-x%7D%3B%5C%20x%3D0%2C1%2C2%2C...)
(a)
Compute the probability that none of the households are tuned to CSI: Shoboygan as follows:
![P(X=0)={14\choose 0}(0.18)^{0}(1-0.18)^{14-0}=1\times1\times0.06214=0.062](https://tex.z-dn.net/?f=P%28X%3D0%29%3D%7B14%5Cchoose%200%7D%280.18%29%5E%7B0%7D%281-0.18%29%5E%7B14-0%7D%3D1%5Ctimes1%5Ctimes0.06214%3D0.062)
Thus, the value of P (None) is 0.062.
(b)
Compute the probability that at least one household is tuned to CSI: Shoboygan as follows:
P (X ≥ 1) = 1 - P (X < 1)
= 1 - P (X = 0)
![=1-0.062\\=0.938](https://tex.z-dn.net/?f=%3D1-0.062%5C%5C%3D0.938)
Thus, the value of P(at least one) is 0.938.
(c)
Compute the probability that at most one household is tuned to CSI: Shoboygan as follows:
P (X ≤ 1) = P (X = 0) + P (X = 1)
![={14\choose 0}(0.18)^{0}(1-0.18)^{14-0}+{14\choose 1}(0.18)^{1}(1-0.18)^{14-1}\\=0.062+0.191\\=0.253](https://tex.z-dn.net/?f=%3D%7B14%5Cchoose%200%7D%280.18%29%5E%7B0%7D%281-0.18%29%5E%7B14-0%7D%2B%7B14%5Cchoose%201%7D%280.18%29%5E%7B1%7D%281-0.18%29%5E%7B14-1%7D%5C%5C%3D0.062%2B0.191%5C%5C%3D0.253)
Thus, the value of P(at most one) is 0.253.
(d)
An event that has a very low probability of occurrence is known as an unusual event.
The probability of the event "at most one household is tuned to CSI: Shoboygan" is 0.253.
This probability value is not low.
Hence, the event is not unusual.