Question 1109794: red, hello and blue counters are placed on a board as shown, and they 'race' to the finish (F) by moving up one square at a time. The moves are determined by picking a bead from random from a bag containing one red bead, two yellow beads and three blue beads. After the colour of the bead which has been drawn is noted the bead is returned to the bag before the next bead is picked. The race is over as soon as one of the counters la do on the square marked F. Find the probability of winning for each counter.
F F F (first line of diagram shows 3 boxes with F for finish in top line in each box)
R * * (Next line of diagram shows 3 boxes with R for red in left hand box and other two blank)
Y * (Next line of diagram shows two boxes with Y for yellow in middle box and RHS one blank)
B (Last line of diagram shows one box with B for blue in it on RHS
Found 3 solutions by ikleyn, Alan3354, MathLover1:
Answer by ikleyn(52848)   (Show Source):
Answer by Alan3354(69443)  (Show Source):
Answer by MathLover1(20850)  (Show Source):
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Blue wins if and only if the sequence of picked beads is one of the following:  ,  ,  , or  .
So the probability that Blue wins is:
Yellow wins if and only if the sequence of picked beads is one of the following:  ,  ,  ,  ,  , or  .
So the probability that Yellow wins is:
Finally Red wins with probability:
As a double-check, we have that Red wins if and only if the sequence of picked beads is one of the following:  ,  ,  ,  , , ,  ,  , or  .
So the probability that Red wins is:
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