Rules of Probability and Mixed Practice

From axioms to work­ing rules#

Once the axioms are in place, the every­day rules for "or" and "not" fol­low almost for free. Add the count­ing you learnt in per­mu­ta­tions and com­bi­na­tions, and ques­tions about cards and com­mit­tees become straight­for­ward prob­a­bil­ity sums. We will prove the addi­tion rule and the com­ple­ment rule, use them on real ques­tions, and end with a mixed prac­tice set on the whole chap­ter.

The basic rules#

Com­ple­ment. $P(A') = 1 - P(A)$. This holds because $A$ and $A'$ are mutu­ally exclu­sive and exhaus­tive. It is often far eas­ier to find the chance of some­thing not hap­pen­ing and sub­tract it from one.

Addi­tion rule. For any two events,

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

Why sub­tract? Because $A \cup B$ is the dis­joint union of $A$ and $B - A$, and $P(B - A) = P(B) - P(A \cap B)$. When the events are mutu­ally exclu­sive, the last term is sim­ply $0$.

"But not". $P(A - B) = P(A) - P(A \cap B)$.

Count­ing with com­bi­na­tions#

Mixed prac­tice on the chap­ter#

1. Two dice are rolled. Find $P(\text{sum} = 9)$, $P(\text{doublet})$, $P(\text{sum} \ge 10)$.
2. Three coins are tossed. Find $P(\text{at least two heads})$ and $P(\text{no tail})$.
3. $P(A) = 0.42$, $P(B) = 0.48$, $P(A \cap B) = 0.16$. Find $P(\text{not } A)$, $P(\text{not } B)$, $P(A \text{ or } B)$, $P(\text{neither})$.
4. In a class of $60$, $30$ play foot­ball, $32$ play cricket and $24$ play both. A stu­dent is cho­sen at ran­dom. Find $P(\text{plays neither})$ and $P(\text{plays exactly one})$.
5. From a group of $6$ boys and $4$ girls, $3$ are cho­sen at ran­dom. Find $P(\text{exactly } 2 \text{ girls})$ and $P(\text{at least } 1 \text{ girl})$.
6. A bag has $5$ red, $4$ blue and $3$ green balls. Two are drawn. Find $P(\text{same colour})$.
7. In a lot­tery a per­son picks $6$ num­bers from $1$ to $20$. Six num­bers are drawn at ran­dom. What is the chance of win­ning, that is, of all six match­ing?
8. $5$ boys and $3$ girls sit in a row at ran­dom. Find $P(\text{the girls are all together})$.
9. Given $\displaystyle P(A) = \tfrac{3}{5}$ and $\displaystyle P(B) = \tfrac{1}{5}$, find $P(A \cup B)$ if $A$ and $B$ are mutu­ally exclu­sive.
10. A num­ber is cho­sen from the $4$-digit num­bers formed by $1, 2, 3, 4, 5$ with­out rep­e­ti­tion. Find $P(\text{it is divisible by } 5)$ and $P(\text{it is even})$.
11. Could $P(A) = 0.6$, $P(B) = 0.5$, $P(A \cap B) = 0.05$ all be true? Explain.
12. Out of $50$ tick­ets num­bered $1$–$50$, one is drawn. Find $P(\text{it is a multiple of } 3 \text{ or a square})$.

Answers to check against#