A die marked $1,\,2,\,3$ in red and $4,\,5,\,6$ in green is tossed. Let $A$ be the event, $'$ the number is even,$'$ and $B$ be the event, 'the number is red'. Are $A$ and $B$ independent?

When a die is thrown, the sample space ( $S$ ) is

$\mathrm{S}=\{1,2,3,4,5,6\}$

Let $A:$ the number is even $=\{2,4,6\}$

$\Rightarrow P(A)=\frac{3}{6}=\frac{1}{2}$

$B:$ the number is red $=\{1,2,3\}$

$\Rightarrow P(B)=\frac{3}{6}=\frac{1}{2}$

$\therefore $ $A \cap B=\{2\}$

$P(A B)=P(A \cap B)=\frac{1}{6}$

$P(A) P(B)=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \neq \frac{1}{6}$

$\Rightarrow $ $P(A) \cdot P(B) \neq P(A B)$

Therefore, $A$ bad $B$ are not independent.

Events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$ and $\mathrm{P}$ $($ not $ \mathrm{A}$ or not $\mathrm{B})=\frac{1}{4} .$ State whether $\mathrm{A}$ and $\mathrm{B}$ are independent?

If $E$ and $F$ are independent events such that $0 < P(E) < 1$ and $0 < P\,(F) < 1,$ then

- [IIT 1989]

The probabilities that a student passes in Mathematics, Physics and Chemistry are $m, p$ and $c$ respectively. On these subjects, the student has a $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly two. Which of the following relations are true

- [IIT 1999]

A die is tossed thrice. Find the probability of getting an odd number at least once.

Three ships $A, B$ and $C$ sail from England to India. If the ratio of their arriving safely are $2 : 5, 3 : 7$ and $6 : 11$ respectively then the probability of all the ships for arriving safely is