A pharmaceutical company gives 10 subjects a new antihistamine, and gives another 10 subjects a placebo (a pill containing no medicine). Afterwards, 8 of the 20 subjects are selected at random to be studied further. Let random variable $X=$ the number of subjects among the 8 people selected who were given the placebo.
a. Find the probability mass function $f(x)=P(X=x)$, and sketch its histogram.
b. Find $P(X=4)$. What does this number represent?
c. Find $P(X \geq 4)$. What does this number represent?
d. Find $P(X \leq 7)$. What does this number represent?
Using the hypergeometric probability distribution model, set up an expression that can be used to find a single ordered pair in the probability mass function $f(x)=P(X=x)$. $f(x)=P(x=x)=\frac{\left(\begin{array}{c}\square \\ x\end{array}\right)\left(\begin{array}{c}\square \\ \square-x\end{array}\right)}{(\square)}$ (Simplify your answers.)
a. Find the probability mass function $f(x)=P(X=x)$.
The probability mass function $f(x)$ is \{\} .
(Type an ordered pair. Use a comma to separate answers as needed. Round to three decimal places as needed.)
Let $x=$ the number of defective pedometers among the 8 units tested, and let $y=f(x)$. Choose the correct histogram below.
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Answer
a. The probability mass function $f(x)=P(X=x)$ can be found using the hypergeometric distribution formula: $f(x)=P(X=x)=\frac{\left(\begin{array}{c}10 \\ x\end{array}\right)\left(\begin{array}{c}10 \\ 8-x\end{array}\right)}{\left(\begin{array}{c}20 \\ 8\end{array}\right)}$ where $x$ is the number of subjects among the 8 people selected who were given the placebo. The possible values of $x$ are 0, 1, 2, ..., 8. b. To find $P(X=4)$, substitute $x=4$ into the formula: $P(X=4)=\frac{\left(\begin{array}{c}10 \\ 4\end{array}\right)\left(\begin{array}{c}10 \\ 4\end{array}\right)}{\left(\begin{array}{c}20 \\ 8\end{array}\right)}$ This number represents the probability that exactly 4 out of the 8 selected subjects were given the placebo. c. To find $P(X \geq 4)$, sum the probabilities $P(X=x)$ for $x=4, 5, ..., 8$. This number represents the probability that at least 4 out of the 8 selected subjects were given the placebo. d. To find $P(X \leq 7)$, sum the probabilities $P(X=x)$ for $x=0, 1, ..., 7$. This number represents the probability that at most 7 out of the 8 selected subjects were given the placebo. The histogram would be a plot of the values of $x$ (from 0 to 8) on the x-axis and the corresponding probabilities $f(x)$ on the y-axis. The shape of the histogram would depend on the specific probabilities calculated.
Step 1 ：a. The probability mass function $f(x)=P(X=x)$ can be found using the hypergeometric distribution formula: $f(x)=P(X=x)=\frac{\left(\begin{array}{c}10 \\ x\end{array}\right)\left(\begin{array}{c}10 \\ 8-x\end{array}\right)}{\left(\begin{array}{c}20 \\ 8\end{array}\right)}$ where $x$ is the number of subjects among the 8 people selected who were given the placebo. The possible values of $x$ are 0, 1, 2, ..., 8. b. To find $P(X=4)$, substitute $x=4$ into the formula: $P(X=4)=\frac{\left(\begin{array}{c}10 \\ 4\end{array}\right)\left(\begin{array}{c}10 \\ 4\end{array}\right)}{\left(\begin{array}{c}20 \\ 8\end{array}\right)}$ This number represents the probability that exactly 4 out of the 8 selected subjects were given the placebo. c. To find $P(X \geq 4)$, sum the probabilities $P(X=x)$ for $x=4, 5, ..., 8$. This number represents the probability that at least 4 out of the 8 selected subjects were given the placebo. d. To find $P(X \leq 7)$, sum the probabilities $P(X=x)$ for $x=0, 1, ..., 7$. This number represents the probability that at most 7 out of the 8 selected subjects were given the placebo. The histogram would be a plot of the values of $x$ (from 0 to 8) on the x-axis and the corresponding probabilities $f(x)$ on the y-axis. The shape of the histogram would depend on the specific probabilities calculated.
