Question

The general addition rule for two events is presented in formula on page 214 and that for three events is displayed on page 216

Part(a) Verify the general addition rule for three events.

Part (b) Write the general addition rule for four events and explain your reasoning.

Short answer

Part (a)

$$\begin{gathered} P(AorBorC)=P\left(A\right)+P(BorC)-P(A\&(BorC\left)\right) \\ P(AorBorC)=P\left(A\right)+P(BorC)-P\left(\right(A\&B)or(A\&C\left)\right) \\ P(AorBorC)=P\left(A\right)+P\left(B\right)+P(C)-P(B\&C)-\left[P\right(A\&B)+P(A\hspace{0.17em}\&C) \\ -P((A\&B)\&(A\&C)\left)\right] \\ P(AorBorC)=P\left(A\right)+P\left(B\right)+P(C)-P(A\&B)-P(B\&C) \\ -P(C\&A)+P(A\&B\&C) \end{gathered}$$

Part (b)

$$\begin{gathered} P(AorBorCorD)=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)-P(A\&B)-P(B\&C)-P(C\&D) \\ -P(A\&D)-P(B\&D)-P(A\&C)+P(B\&C\&D)+P(A\&C\&D)+P(A\&B\&D) \\ -P(A\&B\&C\&D) \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given information.  

There are three events A, B and C

Part (a) Step 2.Using the addition rule in general:

$$\begin{gathered} P(AorBorC)=P\left(A\right)+P(BorC)-P(A\&(BorC\left)\right) \\ P(AorBorC)=P\left(A\right)+P(BorC)-P\left(\right(A\&B)or(A\&C\left)\right) \\ P(AorBorC)=P\left(A\right)+P\left(B\right)+P(C)-P(B\&C)-\left[P\right(A\&B)+P(A\hspace{0.17em}\&C)- \\ P((A\&B)\&(A\&C)\left)\right] \\ P(AorBorC)=P\left(A\right)+P\left(B\right)+P(C)-P(A\&B)-P(B\&C) \\ -P(C\&A)+P(A\&B\&C) \end{gathered}$$

Part (b) Step 1. Given information.  

There are four events A, B ,C and D

Part (b) Step 2.Using the addition rule in general:

$$\begin{gathered} P(AorBorCorD)=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)-P(A\&B)-P(B\&C)-P(C\&D) \\ -P(A\&D)-P(B\&D)-P(A\&C)+P(B\&C\&D)+P(A\&C\&D)+P(A\&B\&D) \\ -P(A\&B\&C\&D) \end{gathered}$$

In order to calculate, we must make sure that the identical results are not counted again. Through a Venn diagram, the expressions for (A or B or C or D) may be simply justified:

Part (b) Step 3.In order to calculate, 

We sum the probabilities of events A, B, C, and D together.

$P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)$.....1

We need to subtract one because we added the chance of intersection twice.

$-P(A\&B)-P(A\&C)-P(A\&D)-P(B\&C)-P(B\&D)-P(C\&D)$......2 Now, we need to add the probability of intersection of three out of four sets three times in (1) and subtract three times in (2), therefore we need to add the probability of intersection of three out of four sets three times in (1) and subtract three times in (2).

$+(A\&B\&C\&)+P(A\&B\&D)+P(A\&C\&D)+P(B\&C\&D)$....3

Now, in (1), we added the probability of intersection of four sets four times, subtracted six times in (2), and added four times in (3), so we only need to remove once more:

$-(P\&B\&C\&D)$ .....4

When we add (1), (2), (3), and (4) together, we get:

$$\begin{gathered} P(AorBorCorD)=P\left(A\right)+P\left(B\right)+P\left(C\right)+P\left(D\right)-P(A\&B)-P(B\&C)-P(C\&D) \\ -P(A\&D)-P(B\&D)-P(A\&C)+P(B\&C\&D)+P(A\&C\&D)+P(A\&B\&D) \\ -P(A\&B\&C\&D) \end{gathered}$$