A slip of paper is given to A, who marks it with either a plus or a minus sign; the probability of his writing a plus is 13. He then passes the slip to B, who may either leave it or change the sign before passing it on to C. Next C passes the slip to D after perhaps changing the sign; finally D passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that B,C and D each change the sign with probability 23. Then probability that A originally wrote a plus is ab (where a,b are coprime numbers), then the value of a+b is

A slip of paper is given to A, who marks it with either a plus or a mi

1.	A slip of paper is given to A, who marks it with either a plus or a minus sign; the probability of his writing a plus is 13. He then passes the slip to B, who may either leave it or change the sign before passing it on to C. Next C passes the slip to D after perhaps changing the sign; finally D passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that B,C and D each change the sign with probability 23. Then probability that A originally wrote a plus is ab (where a,b are coprime numbers), then the value of a+b is
Answer» A slip of paper is given to $A$ , who marks it with either a plus or a minus sign; the probability of his writing a plus is $\frac{1}{3}$ . He then passes the slip to $B$ , who may either leave it or change the sign before passing it on to $C$ . Next $C$ passes the slip to $D$ after perhaps changing the sign; finally $D$ passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that $B, C$ and $D$ each change the sign with probability $\frac{2}{3}$ . Then probability that $A$ originally wrote a plus is $\frac{a}{b}$ (where $a, b$ are coprime numbers), then the value of $a + b$ is