Two independent events each have a probability of 73 percent. What is the combined probability?

Step-by-step solution

1. 1

   Notice that the question is ambiguous, and say so. "Combined" can mean both events happen or at least one happens. These are different numbers, and the gap here is nearly $40$ percentage points — so the interpretation has to be settled before any arithmetic. One thing it never means is adding the two: $73\%+73\%=146\%$ is impossible, since probabilities cannot exceed $1$.

2. 2

   Interpretation 1 - both events occur. For independent events the probabilities multiply:

   $P(A\text{ and }B)=P(A)\times P(B)=0.73\times0.73=0.5329=53.29\%$

   Requiring two things at once is always harder than either alone, so the answer must be below $73\%$ ✓.

3. 3

   Interpretation 2 - at least one occurs. Use the complement: "at least one" fails only when both fail, and each fails with probability $1-0.73=0.27$:

   $P(\text{neither})=0.27\times0.27=0.0729$

   $P(\text{at least one})=1-0.0729=0.9271=92.71\%$

   Requiring only one of two chances is easier than either alone, so this must exceed $73\%$ ✓.

4. 4

   Check the two answers against each other. The four disjoint outcomes must sum to $1$: both $=0.5329$, only the first $=0.73\times0.27=0.1971$, only the second $=0.1971$, neither $=0.0729$. Adding: $0.5329+0.1971+0.1971+0.0729=1.0000$ ✓. This also shows exactly one occurs with probability $0.3942=39.42\%$.

5. 5

   Note where independence matters. Both formulas assume the events do not influence each other. If they were perfectly correlated, "both" would be $73\%$ and "at least one" would also be $73\%$; if mutually exclusive, "both" would be $0$. Independence is an assumption to state, not a default.

6. 6

   Summarise the two answers. Both occur: $\mathbf{53.29\%}$. At least one occurs: $\mathbf{92.71\%}$. Every figure above was recomputed and the four outcomes verified to sum to exactly $1$ ✓.