The Range of Probability and Complementary Events

Probability Never Leaves the Range 0 to 1

Because the number of favourable outcomes can be at least 0 and at most the total, every probability satisfies $$0 \le P(E) \le 1.$$

• $P(E) = 0$ means $E$ cannot happen.
• $P(E) = 1$ means $E$ is certain.
• A value in between measures partial likelihood ($\dfrac12$ is an even chance).

Key Point: A probability can never be negative and never exceed 1. If your answer is $-\dfrac13$ or $\dfrac{7}{6}$, you have miscounted.

Sure Events and Impossible Events

• A sure (certain) event always happens — every outcome is favourable — so $P = \dfrac{\text{total}}{\text{total}} = 1$. Example: "getting a number from 1 to 6" when throwing a die.
• An impossible event never happens — no outcome is favourable — so $P = \dfrac{0}{\text{total}} = 0$. Example: "getting 8" on an ordinary die.

[Board Important] "Certain / sure" $\Rightarrow P=1$; "impossible / cannot happen" $\Rightarrow P=0$.

Complementary Events

For any event $E$, the event "$E$ does not happen" is its complement, written $\overline{E}$ (or "not $E$"). Between them they cover every outcome, so their probabilities add to 1: $$\boxed{P(\overline{E}) = 1 - P(E)}$$

This is one of the most useful shortcuts in the chapter: if the event you want is awkward to count, count its complement instead and subtract from 1.

Key Point: $P(E) + P(\overline{E}) = 1$. "At least one", "not", "none" problems are often easiest via the complement.

Using the Complement

Watch for wording that begs for the complement:

• "the probability that it is not a king" $= 1 - P(\text{king})$,
• "the probability of getting at least one head" $= 1 - P(\text{no head})$.

Counting the complement is often far quicker than listing every favourable case.

Key Point: When "not / at least / none" appears, ask: is $P(\overline{E})$ easier to find? Then use $P(E)=1-P(\overline{E})$.

Solved Examples

Example 1: Complement on a die

A die is thrown. Find the probability of not getting a 6.

Solution:

1. $P(6) = \dfrac16$.
2. $P(\text{not }6) = 1 - \dfrac16 = \dfrac56$.

Final Answer: $\dfrac56$.

Takeaway: $P(\overline{E}) = 1 - P(E)$.

Example 2: Impossible and sure events

A die is thrown. Find the probability of (i) getting 8, (ii) getting a number less than 7.

Solution:

1. (i) No face shows 8 $\Rightarrow$ impossible $\Rightarrow P = 0$.
2. (ii) Every face ($1$–$6$) is less than 7 $\Rightarrow$ sure $\Rightarrow P = 1$.

Final Answer: (i) $0$; (ii) $1$.

Takeaway: Impossible $\to 0$, certain $\to 1$.

Example 3: Find the complement's probability

The probability that it will rain tomorrow is $0.85$. What is the probability that it will not rain?

Solution:

1. $P(\text{no rain}) = 1 - 0.85 = 0.15$.

Final Answer: $0.15$.

Takeaway: Complementary probabilities add to 1.

Example 4: Is this a valid probability?

Can the probability of an event be $1.4$? What about $-0.2$?

Solution:

1. Probabilities must satisfy $0 \le P \le 1$.
2. Both $1.4$ and $-0.2$ lie outside $[0,1]$, so neither can be a probability.

Final Answer: No — a probability cannot exceed 1 or be negative.

Takeaway: Always check that your answer is between 0 and 1.