Question

Complements and the Addition Rule Refer to the table used for Exercises 9–20. Assume that one order is randomly selected. Let A represent the event of getting an order from McDonald’s and let B represent the event of getting an order from Burger King. Find $P\left(\overline{AorB}\right)$, find $P\left(\overline{A}or\overline{B}\right)$, and then compare the results. In general, does $P\left(\overline{AorB}\right)$= $P\left(\overline{A}or\overline{B}\right)$?

Short answer

$P\left(\overline{AorB}\right)$is equal to 0.392.

$P\left(\overline{A}or\overline{B}\right)$is approximately equal to 1.

No, the two probabilities are not equal.

Step-by-step solution

Given information

The number of food orders from four fast-food chains is provided.

The events are defined as follows:

A: getting an order from McDonald’s and B: getting an order from Burger King.

Addition rule of probability 

The addition rule has the following expression:

$P\left(AorB\right)=P\left(A\right)+P\left(B\right)-P\left(AandB\right)$

This represents the probability of occurrence of either A or B, or both.

Complementary events

A and $\overline{A}$are said to be complementary events when the former signifies the occurrence of an event while the latter is the non-occurrence of the same event.

$P\left(A\right)+P\left(\overline{A}\right)=1$

Compute the probability

The following table is considered for all calculations (as in Exercise 9 to 20)

	McDonald’s	Burger King	Wendy’s	Taco Bell	Totals
Order Accurate	329	264	249	145	987
OrderNotAccurate	33	54	31	13	131
Totals	362	318	280	158	1118

Let A be the event of getting a food order from McDonald’s.

Let B be the event of getting a food order from Burger King.

The probability of getting a food order from McDonald’s is equal to:

$P\left(A\right)=\frac{362}{1118}$

The probability of getting a food order from Burger King is equal to:

$P\left(B\right)=\frac{318}{1118}$

The number of food orders from Burger King as well as McDonald's is equal to 0.

The probability of getting a food order from McDonald’s and Burger King is given as:

$$\begin{gathered} P\left(AandB\right)=\frac{0}{1118} \\ =0 \end{gathered}$$

Using the complement rule, the required probability is expressed as follows:

$$\begin{gathered} P\left(\overline{AorB}\right)=1-P\left(AorB\right) \\ =1-\left[P\left(A\right)+P\left(B\right)-P\left(AorB\right)\right] \\ =1-\left[\frac{362}{1118}+\frac{318}{1118}-\frac{0}{1118}\right] \\ =0.392 \end{gathered}$$

Therefore,$P\left(\overline{AorB}\right)=0.392$

Compute the probability

The probability of not getting a food order from McDonald’s is equal to:

$$\begin{gathered} P\left(\overline{A}\right)=1-\frac{362}{1118} \\ =\frac{756}{1118} \end{gathered}$$

The probability of not getting a food order from Burger King is equal to:

$$\begin{gathered} P\left(\overline{B}\right)=1-\frac{318}{1118} \\ =\frac{800}{1118} \end{gathered}$$

The number of food orders that are neither from McDonald’s nor Burger King is equal to the sum of orders from the remaining two chains.

$280+158=438$

The probability of getting a food order neither from McDonald’s nor Burger King is equal to:

$$\begin{gathered} P\left(\overline{A}or\overline{B}\right)=P\left(\overline{A}\right)+P\left(\overline{B}\right)-P\left(\overline{A}and\overline{B}\right) \\ =\frac{756}{1118}+\frac{800}{1118}-\frac{438}{1118} \\ =1 \end{gathered}$$

Therefore, $P\left(\overline{A}or\overline{B}\right)\approx 1$.

Compare the two probabilities

The numerical values of the two probabilities are unequal.

Thus, it cannot be concluded for the general setup that $P\left(\overline{AorB}\right)=P\left(\overline{A}or\overline{B}\right)$.