Consider a system of equations $ax + by = 0$ and $cx + dy = 0$,where $a, b, c, d \in \{0, 1\}$.
Statement $-1$: The probability that the system of equations has a solution is $1$.
Statement $-2$: The probability that the system of equations has a unique solution is $\frac{3}{8}$.
Statement $-1$: The probability that the system of equations has a solution is $1$.
Statement $-2$: The probability that the system of equations has a unique solution is $\frac{3}{8}$.
- AStatement $-1$ is true,Statement $-2$ is true,but Statement $-1$ is not the correct explanation for Statement $-2$.
- BStatement $-1$ is true,Statement $-2$ is false.
- CStatement $-1$ is false,Statement $-2$ is true.
- DBoth statements are true,and Statement $-1$ is the correct explanation of Statement $-2$.
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Three screws are drawn at random from a lot of $50$ screws containing $5$ defective ones. The probability of the event that all $3$ screws drawn are non-defective,assuming that the drawing is $(a)$ with replacement and $(b)$ without replacement,is respectively:
$A$ random variable $X$ has the following probability distribution:
| $X=x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $P(X=x)$ | $0.15$ | $0.23$ | $0.12$ | $0.20$ | $0.08$ | $0.10$ | $0.05$ | $0.07$ |
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 5\}$,find $P(E \cup F)$.
| $X=x$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $P(X=x)$ | $0.15$ | $0.23$ | $0.12$ | $0.20$ | $0.08$ | $0.10$ | $0.05$ | $0.07$ |
For the events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 5\}$,find $P(E \cup F)$.
$E_1$ and $E_2$ are two independent events of a random experiment with $P(E_1) = \frac{1}{2}$ and $P(E_1 \cup E_2) = \frac{2}{3}$. Match the items of List-$I$ with those of List-$II$.
List-$I$ List-$II$ $A. P(E_2) =$ $I. 2/3$ $B. P(E_1 | E_2) =$ $II. 5/6$ $C. P(\bar{E}_2 | E_1) =$ $III. 1/3$ $D. P(\bar{E}_1 \cup \bar{E}_2) =$ $IV. 1/2$
| List-$I$ | List-$II$ |
| $A. P(E_2) =$ | $I. 2/3$ |
| $B. P(E_1 | E_2) =$ | $II. 5/6$ |
| $C. P(\bar{E}_2 | E_1) =$ | $III. 1/3$ |
| $D. P(\bar{E}_1 \cup \bar{E}_2) =$ | $IV. 1/2$ |
Let $X$ and $Y$ be events such that $P(X \cup Y) = P(X \cap Y).$
Statement-$1$: $P(X \cap Y) = P(X' \cap Y') = 0$
Statement-$2$: $P(X) + P(Y) = 2P(X \cap Y).$
Statement-$1$: $P(X \cap Y) = P(X' \cap Y') = 0$
Statement-$2$: $P(X) + P(Y) = 2P(X \cap Y).$
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