The addition rule

The 2 × 2 layout

Two events on one population sort every outcome into four cells. For Maple Ledger's invoices:

              error   no error   total
  late          3         9        12
  on time       5        33        38
  total         8        42        50

The three "both" invoices fill the late-and-error cell; the row and column totals follow (12 − 3 = 9 late without an error; 8 − 3 = 5 on-time with an error; 50 − 17 = 33 neither). Fill the margins first — every later probability is a cell or a margin divided by something.

The general addition rule

"Late or error" is the L-shaped set of three cells: $3+9+5=17$. Adding the two margins, $12+8=20$, counts the corner cell twice, so subtract it once:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

$$P(\text{late}\cup \text{error})=0.24+0.16-0.06=0.34$$

A Venn diagram says the same thing: two overlapping circles, the lens counted in both, so it is taken out once. The rule is not optional — $P(A) + P(B)$ alone is wrong whenever the events can happen together.

Mutually exclusive events

Two events are mutually exclusive (disjoint) when they cannot both happen: $P(A\cap B)=0$. Then the overlap term vanishes and

$$P(A\cup B)=P(A)+P(B)\text{(mutually exclusive only)}$$

Georgian Bay Mutual prices a policy where, in a given year, a claim is either a total loss (probability 0.30) or a partial loss (0.25), never both. $P(\text{a loss})=0.30+0.25=0.55$. The special case is the general rule with a zero plugged in — learn the general one, and check whether the overlap is zero before dropping it.

The four cells sum to one

As probabilities, the table is 0.06, 0.18, 0.10, 0.66 — summing to 1.00, as the four cells must, because every invoice lands in exactly one of them. That check catches most arithmetic slips before they reach the answer line.

What "or" does not mean

In everyday speech "late or error" can mean one but not both. In probability it never does; "exactly one of them" is a different event, $0.18+0.10=0.28$, and the paper says so explicitly when it wants it.