$A$ six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be $7$. The probability that the number $3$ has appeared at least once is:

If a dice is thrown twice,then the probability of getting $1$ in the first throw only is

$A$ manager decides to distribute $Rs. 20000$ between two employees $X$ and $Y$. He knows $X$ deserves more than $Y$,but does not know how much more. So,he decides to arbitrarily break $Rs. 20000$ into two parts and gives $X$ the bigger part. Then,the chance that $X$ gets twice as much as $Y$ or more is

Let $A$ and $B$ be two independent events. The probability that both $A$ and $B$ happen is $\frac{1}{12}$ and the probability that neither $A$ nor $B$ happens is $\frac{1}{2}$. Then

$A$ coin and a six-faced die,both unbiased,are thrown simultaneously. The probability of getting a head on the coin and an odd number on the die is:

If $A$ and $B$ are two events such that $P(A \cup B) = P(A \cap B)$,then the incorrect statement amongst the following statements is