Ultimate Binomial Exercises! Exercises 37–40 involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities.

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning line positions on voting ballots. Among 41 different ballots, Democrats were assigned the top line 40 times. Assume that Democrats and Republicans are assigned the top line using a method of random selection so that they are equally likely to get that top line.

a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 40 top lines for Democrats significantly high?

b. Find the probability of exactly 40 top lines for Democrats

c. Find the probability of 40 or more top lines for Democrats.

d. Which probability is relevant for determining whether 40 top lines for Democrats is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 40 top lines for Democrats significantly high?

e. What do the results suggest about how the clerk met the requirement of assigning the line positions using a random method?

It is given that out of 41 voting ballots, Democrats were assigned the top line 40 times.

Democrats and Republicans are equally likely to get the top line.

Here,

Let X denote the number of top lines for democrats.

Success is defined as getting a Democrat who was assigned the top line.

It is given to assume that Democrats and Republicans are equally likely to get that top line.

The probability of success (Democrat) is equal to the probability of failure (Republican):

$p=0.5$

The total number of ballots (n) is equal to 41.

The probability of selecting a Democrat who was assigned the top line is given as

p=0.5.

The mean of the number of Democrats who were assigned the top line is as follows:

$$\begin{gathered} \mu =np \\ =\left(41\right)\left(0.5\right) \\ =20.5 \end{gathered}$$

Thus, $\mu =20.5$.

The standard deviation for the number of green candies is computed below:

$$\begin{gathered} \sigma =\sqrt{npq} \\ =\sqrt{41\times 0.5\times 0.5} \\ =3.20 \end{gathered}$$

Thus, $\sigma =3.20$.

a.

The following limit separates significantly low values:

$$\begin{gathered} \mu -2\sigma =20.5-\left(2\right)\left(3.20\right) \\ =14.1 \end{gathered}$$

Therefore, values less than or equal to 14.10 are considered to be significantly low.

The following limit separates significantly high values:

$$\begin{gathered} \mu +2\sigma =20.5+\left(2\right)\left(3.20\right) \\ =26.9 \end{gathered}$$

Therefore, values greater than or equal to 26.9 are considered to be significantly high.

Values lying between 14.1 and 26.9 are not significant.

Here, the value of 40 is greater than 26.9.

Thus, the value of 40 is considered significantly high.

b.

The binomial probability formula used to compute the given probability is as follows:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

(The symbols have been defined in Step 1).

Using the binomial probability formula, the probability of getting exactly 40 top lines for Democrats is as follows:

$$\begin{gathered} P\left(X=40\right)={}^{41}{C}_{40}{\left(0.5\right)}^{40}{\left(0.5\right)}^{41-40} \\ =0.000000000019 \end{gathered}$$

Thus, the probability of getting exactly 40 top lines for Democrats is equal to 0.000000000019.

c.

The probability of getting 40 or more top lines for Democrats has the following expression:

$$\begin{gathered} P\left(X⩾40\right)=P\left(X=40\right)+P\left(X=41\right) \\ ={}^{41}{C}_{40}{\left(0.5\right)}^{40}{\left(0.5\right)}^1+{}^{41}{C}_{41}{\left(0.5\right)}^{41}{\left(0.5\right)}^0 \\ =0.000000000019099 \end{gathered}$$

Thus, the probability of getting 40 or more top lines for Democrats is equal to 0.000000000019099.

d.

The following is the probability formula for determining whether or not a given value of the number of successes (x) is significantly high:

$P\left(x\text{or}\text{more}\right)⩽0.05$

Here, the considered value of x is equal to 40.

Since the probability of 40 or more top lines for Democrats is represented in part (c), theprobability computed in part (c) is relevant for determining whether the result of 40 top lines for Democrats is significantly high.

Thus,

$$\begin{gathered} P\left(40\text{or}\text{more}\right)=0.000000000019099 \\ <0.05 \end{gathered}$$

Since the probability of 40 or more top lines for Democrats is less than 0.05, the value of 40 top lines for Democrats is significantly high.

e.

It is proposed that the clerk had randomly assigned 40 top-line positions to Democrats.

Since the probability of 40 top lines for Democrats is very small, indicating that the event of 40 top lines for Democrats out of 41 ballots is highly unlikely.

It implies that the line positions do not seem to be assigned randomly.

Thus, the clerk did not meet the requirements of assigning the line positions using a random method.