Question

Rule of Complements When randomly selecting an adult, let B represent the event of randomly selecting someone with type B blood. Write a sentence describing what the rule of complements is telling us:$P\left(Bor\overline{B}\right)=1$

Short answer

$P\left(Bor\overline{B}\right)=1$tells that either a randomly selected person will have blood type B or they will not have blood type B.

In other words, one of the two events B and$\overline{B}$ will surely occur.

Step-by-step solution

Given information

Let B be the event of selecting a person with type B blood.

Rule of probability for complementary events

If A represents the occurrence of an event and $\overline{A}$represents the occurrence of “not A,” then A and$\overline{A}$ arecomplementary events.

The outcomes that do not belong to event A belong to the complementary set and vice-versa. Consequently, only one of the two events can occur at one point in time.

Moreover, one of the two events is sure to occur.

Mathematically,

$$\begin{gathered} P\left(Aor\overline{A}\right)=P\left(A\right)+P\left(\overline{A}\right) \\ =1 \end{gathered}$$

In terms of given event B

Here, B is the event of selecting a person with type B blood. Then, is the event of selecting a person whose blood type is not B.

Thus, the probability that either of the two events will occur is given as:

$$\begin{gathered} P\left(Bor\overline{B}\right)=P\left(B\right)+P\left(\overline{B}\right)-P\left(Band\overline{B}\right)...\left(\text{Addition}\text{rule}\right) \\ =P\left(B\right)+P\left(\overline{B}\right)-0 \\ =P\left(B\right)+P\left(\overline{B}\right) \\ =1 \end{gathered}$$

This implies that events B and$\overline{B}$ cannot occur together, and either of the two events will certainly occur.

Specifically, the probability that a person either has blood type B or does not have blood type B is sure to occur for any randomly selected adult.