Question 1109794

Blue wins if and only if the sequence of picked beads is one of the following: {{{BBB}}}, {{{YBBB}}}, {{{BYBB}}}, or {{{BBYB}}}. 

So the probability that Blue wins is:

{{{(1+3/3)*(1/2^3)=1/4}}}

Yellow wins if and only if the sequence of picked beads is one of the following: {{{YY}}}, {{{YBY}}}, {{{BYY}}}, {{{YBBY}}}, {{{BYBY}}}, or {{{BBYY}}}. 

So the probability that Yellow wins is:

{{{(1+2/2+3/2^2)*(1/3^2)=11/36}}}

Finally Red wins with probability:

{{{1-1/4 -11/36= 4/9}}}

As a double-check, we have that Red wins if and only if the sequence of picked beads is one of the following: {{{R}}}, {{{YR}}}, {{{BR}}}, {{{YBR}}},{{{ BYR}}},{{{ BBR}}}, {{{YBBR}}}, {{{BYBR}}}, or {{{BBYR}}}. 

So the probability that Red wins is:

{{{(1+1/2+1/3+1/2^2+2/(2*3)+3/(2^2*3))*16=4/9}}}