Q.Two dice are thrown together and the total score is noted. The events $E$, $F$ and $G$ are 'a total of $4$', 'a total of $9$ or more', and 'a total divisible by $5$', respectively. Calculate $P(E)$, $P(F)$ and $P(G)$ and decide which pairs of events, if any, are independent.

Concept: Event Independence — two events are independent iff $P(A \cap B) = P(A) \cdot P(B)$.

Step 1: Total outcomes

Two dice give $6 \times 6 = 36$ equally likely outcomes.

Step 2: Compute probabilities

• $E$ (total 4): outcomes $(1,3), (2,2), (3,1)$ → 3 outcomes. $P(E) = \frac{3}{36} = \frac{1}{12}$.
• $F$ (total 9 or more): totals 9,10,11,12. Total 9: 4 outcomes; 10: 3; 11: 2; 12: 1 → total $4+3+2+1=10$ outcomes. $P(F) = \frac{10}{36} = \frac{5}{18}$.
• $G$ (total divisible by 5): totals 5 and 10. Total 5: 4 outcomes; total 10: 3 outcomes → total 7 outcomes. $P(G) = \frac{7}{36}$.

Step 3: Check independence for each pair

• $E \cap F$: total 4 and total ≥9 → impossible. $P(E \cap F) = 0 \neq \frac{1}{12} \cdot \frac{5}{18}$. Not independent. …

Throwing two dice gives $6\times6=36$ equally likely ordered outcomes. The events are $E:$ total $=4$, $F:$ total $\ge 9$, $G:$ total divisible by $5$ (i.e. total $5$ or $10$).

1. $P(E)$. Total $4$: $(1,3),(2,2),(3,1)$ — $3$ outcomes.

2. $P(F)$. Totals $9,10,11,12$: $4+3+2+1=10$ outcomes.

3. $P(G)$. Total $5$: $4$ outcomes; total $10$: $3$ outcomes; $4+3=7$.

4. Independence tests.

• $E$ and $F$: no total is both $4$ and $\ge 9$, so $E\cap F=\varnothing$ and $P(E\cap F)=0$, while $P(E)P(F)=\dfrac{1}{12}\cdot\dfrac{5}{18}=\dfrac{5}{216}\neq0$. Not independent.
• $E$ and $G$: $4$ is not divisible by $5$, so $E\cap G=\varnothing$ and $P(E\cap G)=0$, while $P(E)P(G)=\dfrac{1}{12}\cdot\dfrac{7}{36}=\dfrac{7}{432}\neq0$. Not independent. …

Method: Computing Event Probabilities and Testing Pairwise Independence

Use this when several events are defined on one sample space and you must find their probabilities and check which pairs are independent.

Steps

Step 1: Count each event over the full sample space.

For two dice there are $36$ equally likely ordered outcomes. For each event list the qualifying outcomes and divide by $36$.

Step 2: For each pair, find the intersection by listing shared outcomes.

$P(A\cap B)$ comes from outcomes that satisfy both conditions — never from multiplying (that is the very thing being tested).

Step 3: Apply the product-rule test to every pair. …

Common Mistakes

Mistake 1: Concluding $E$ and $F$ are independent because $P(E\cap F)=0$.

Why it's wrong: a total cannot be both $4$ and $\ge9$, so they are mutually exclusive — and mutually exclusive events with non-zero probabilities are dependent, since $P(E)P(F)\ne0$. Correct approach: test $P(E\cap F)=P(E)P(F)$; $0\ne\frac{5}{216}$.

Mistake 2: Computing $P(F\cap G)$ by multiplying instead of listing. …