The television show NBC Sunday Night Football broadcast a game between the Colts and Patriots and received a share of 22, meaning that among the TV sets in use, 22% were tuned to that game (based on data from Nielsen Media Research). An advertiser wants to obtain a second opinion by conducting its own survey, and a pilot survey begins with 20 households having TV sets in use at the time of that same NBC Sunday Night Football broadcast.
a. Find the probability that none of the households are tuned to NBC Sunday Night Football.
b. Find the probability that at least one household is tuned to NBC Sunday Night Football.
c. Find the probability that at most one household is tuned to NBC Sunday Night Football.
d. If at most one household is tuned to NBC Sunday Night Football, does it appear that the 22% share value is wrong? Why or why not?
Solution
Here $X$ denote the number of Household having TV sets in use at the time of that same NBC Sunday Night Football broadcast.
Let $p$ be the probability of the TV sets in use, 22% were tuned to that game .
Given that $p=0.22$ and $n =20$. Thus$X\sim B(20, 0.22)$.
The probability mass function of $X$ is
$$ \begin{aligned} P(X=x) &= \binom{20}{x} (0.22)^x (1-0.22)^{20-x},\\ &\quad \; x=0,1,\cdots, 20. \end{aligned} $$
a. The probability that none of the households are tuned to NBC Sunday Night Football is
$$ \begin{aligned} P(X= 0) & =\binom{20}{0} (0.22)^{0} (1-0.22)^{20-0}\\\\ & = 0.0069\\ \end{aligned} $$
b. The probability that at least one household is tuned to NBC Sunday Night Football is
$$ \begin{aligned} P(X\geq 1) & =1- P(X=0)\\ &=1- \binom{20}{0} (0.22)^0 (1-0.22)^{20-0}\\ &= 1-0.0069\\ &= 0.9931 \end{aligned} $$
c. The probability that at most one household is tuned to NBC Sunday Night Football is
$$ \begin{aligned} P(X\leq 1) & =P(X=0)+P(X=1)\\ &= \binom{20}{0} (0.22)^0 (1-0.22)^{20-0}+\binom{20}{1} (0.22)^1 (1-0.22)^{20-1}\\ &= 0.0069+0.0392\\ &= 0.0461 \end{aligned} $$
d. The probability that at most one household is 0.0461. It is less than 0.05 so it appears that the share is not 22.
Conclusion
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