Question

binomial tables handout - 2025 - statistics - mr. scheel

1. what are the characteristics (or properties) that define a binomial random variable?

for problems 2 through 6, complete each binomial table and verify that it is valid (i.e. probabilities sum to 1):

1. for every day during a week(7 days) span, there is a 30% chance of rain. let x = the number of days that it rains.
2. a student takes a 5 - question multiple - choice quiz and guesses every single question. they have a 25% chance of getting any question right. let x = the number of questions answered correctly.

Explanation:

Step1: Recall binomial random - variable properties

A binomial random variable has the following properties: 1. Fixed number of trials ($n$). 2. Each trial has only two possible outcomes (success or failure). 3. The probability of success ($p$) is constant for each trial. 4. The trials are independent of each other.

Step2: Solve problem 2

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n = 7$ (number of days), $p=0.3$ (probability of rain), and $k$ ranges from $0$ to $7$.
For $k = 0$:
\[

$$\begin{align*} P(X = 0)&=C(7,0)\times(0.3)^{0}\times(0.7)^{7}\\ &=1\times1\times0.0823543\\ &=0.0823543 \end{align*}$$

\]
For $k = 1$:
\[

$$\begin{align*} P(X = 1)&=C(7,1)\times(0.3)^{1}\times(0.7)^{6}\\ &=7\times0.3\times0.117649\\ &=0.2470629 \end{align*}$$

\]
Continuing this way for $k$ from $0$ to $7$ and then summing the probabilities: $\sum_{k = 0}^{7}P(X = k)=1$.

Step3: Solve problem 3

Here, $n = 5$ (number of questions), $p = 0.25$ (probability of getting a question right), and $k$ ranges from $0$ to $5$.
For $k = 0$:
\[

$$\begin{align*} P(X = 0)&=C(5,0)\times(0.25)^{0}\times(0.75)^{5}\\ &=1\times1\times0.2373047\\ &=0.2373047 \end{align*}$$

\]
For $k = 1$:
\[

$$\begin{align*} P(X = 1)&=C(5,1)\times(0.25)^{1}\times(0.75)^{4}\\ &=5\times0.25\times0.31640625\\ &=0.3955078125 \end{align*}$$

\]
Continuing for $k$ from $0$ to $5$ and summing the probabilities: $\sum_{k = 0}^{5}P(X = k)=1$.

Answer:

1. The characteristics of a binomial random variable are: fixed number of trials ($n$), two - possible outcomes per trial, constant probability of success ($p$) across trials, and independent trials.
2. The binomial probabilities for $X$ (number of rainy days) where $n = 7$ and $p=0.3$ are calculated as shown above and the sum of probabilities is $1$.
3. The binomial probabilities for $X$ (number of correct questions) where $n = 5$ and $p = 0.25$ are calculated as shown above and the sum of probabilities is $1$.