$A$ bag contains $n + 1$ coins. Among these coins,one coin has heads on both sides,while all other coins are fair. $A$ coin is chosen at random from the bag. If the probability of getting a head when the chosen coin is tossed is $7/12$,what is the value of $n$?
- A$3$
- B$4$
- C$5$
- DNone of these
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$A$ and $B$ throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. $A$ wins if he throws $6$ before $B$ throws $7$,and $B$ wins if he throws $7$ before $A$ throws $6$. If $A$ begins,the probability of $A$ winning is:
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The corners of regular tetrahedrons are numbered $1, 2, 3, 4$. Three tetrahedrons are tossed. The probability that the sum of the upward corners will be $5$ is
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View Solution$S$ is the sample space and $A, B$ are two events of a random experiment. Match the items of List-$A$ with the items of List-$B$.
List-$A$ List-$B$ $(I)$ $A, B$ are mutually exclusive events $(i)$ $P(A \cap B) = P(B) - P(\bar{A})$ $(II)$ $A, B$ are independent events $(ii)$ $P(A) \leq P(B)$ $(III)$ $A \cap B = A$ $(iii)$ $P(\frac{\bar{A}}{B}) = 1 - P(A)$ $(IV)$ $A \cup B = S$ $(iv)$ $P(A \cup B) = P(A) + P(B)$ $(v)$ $P(A) + P(B) = 2$
| List-$A$ | List-$B$ |
|---|---|
| $(I)$ $A, B$ are mutually exclusive events | $(i)$ $P(A \cap B) = P(B) - P(\bar{A})$ |
| $(II)$ $A, B$ are independent events | $(ii)$ $P(A) \leq P(B)$ |
| $(III)$ $A \cap B = A$ | $(iii)$ $P(\frac{\bar{A}}{B}) = 1 - P(A)$ |
| $(IV)$ $A \cup B = S$ | $(iv)$ $P(A \cup B) = P(A) + P(B)$ |
| $(v)$ $P(A) + P(B) = 2$ |
$A$ pair of $12$-sided fair dice with faces numbered $1, 2, 3, \ldots, 12$ is rolled. The probability that the sum of the numbers appearing has a remainder of $2$ when divided by $9$ is
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