Let $A$ and $B$ be independent events such that $P(A)=p$ and $P(B)=2p$. The largest value of $p$, for which $P(\text{exactly one of } A, B \text{ occurs}) = \frac{5}{9}$, is:

The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$,respectively. If $A$ is selected as the captain,the probability that the team wins the tournament is $0.8$ and if $B$ is selected as the captain,the probability that the team wins the tournament is $0.7$. Then the probability,that the team wins the tournament,is:

If $A$ and $B$ are independent events of a random experiment such that $P(A \cap B) = \frac{1}{6}$ and $P(\bar{A} \cap \bar{B}) = \frac{1}{3}$,then $P(A)$ is equal to (Here,$\bar{E}$ is the complement of the event $E$)

Let a computer program generate only the digits $0$ and $1$ to form a string of binary numbers. The probability of occurrence of $0$ at even places is $\frac{1}{2}$ and the probability of occurrence of $0$ at odd places is $\frac{1}{3}$. Then the probability that $'10'$ is followed by $'01'$ is equal to:

There are three bags $B_1$,$B_2$,and $B_3$ containing $2$ Red and $3$ White,$5$ Red and $5$ White,and $3$ Red and $2$ White balls respectively. $A$ ball is drawn from bag $B_1$ and placed in bag $B_2$,then a ball is drawn from bag $B_2$ and placed in bag $B_3$,then a ball is drawn from bag $B_3$. The number of ways in which this process can be completed,if the same colour balls are used in the first and second transfers (assume all balls to be distinct),is

If four positive integers are selected randomly from the set of positive integers,then the probability that the unit digit of their product is $1, 3, 7,$ or $9$ is: