$A$ signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively,is received by station $A$ and then transmitted to station $B$. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station $B$ is green,then the probability that the original signal was green is

Two digits are selected at random from the digits $1$ through $9$. If their sum is even,then the probability that both are odd is

$A$ pack of playing cards was found to contain only $51$ cards. If the first $13$ cards which are examined are all red,then the probability that the missing card is black,is

There are three families $F_1, F_2, F_3$. $F_1$ has $2$ boys and $1$ girl; $F_2$ has $1$ boy and $2$ girls; $F_3$ has $1$ boy and $1$ girl. $A$ family is randomly chosen and a child is chosen from that family randomly. If it is known that the child thus selected is a girl,then the probability that she is from $F_2$ is

$A$ box contains $4$ black,$2$ white,and $6$ red balls. Another box contains $3$ black and $5$ white balls. An unbiased die is thrown. If $1$ or $2$ appears on the die,a ball is drawn from the first box; otherwise,a ball is drawn from the second box. If the drawn ball is black,what is the probability that $2$ appeared on the die?

$A$ company has two plants $A$ and $B$ to manufacture motorcycles. $60 \%$ of motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. $80 \%$ of the motorcycles manufactured at plant $A$ are rated of standard quality,while $90 \%$ of the motorcycles manufactured at plant $B$ are rated of standard quality. $A$ motorcycle picked up randomly from the total production is found to be of standard quality. If $p$ is the probability that it was manufactured at plant $B$,then $126 p$ is