Worked solution
Account for the first red ball being removed
A funfair player draws coloured balls from a box containing five blue and two red balls. To win, the player must pick two red balls consecutively. Each blue ball drawn is replaced and drawing continues; each red ball is not replaced. Calculate the probability of winning. marks
[2014/SRJC/II/5(c)]
CJC · Review problems · Question 7
Answer
.
Describe every possible winning path
A winning sequence is some number of initial blue draws followed immediately by the two red draws. Once a red has been followed by a blue, the removed first red cannot form a later consecutive pair; only one red remains. Before the first red, a blue has probability 5/7. The two final reds have probabilities 2/7 then 1/6.
Retain the source winning-draw table
| Case | Description | Probability |
|---|---|---|
| 1 | Win on draw 2 | |
| 2 | Win on draw 3 | |
| 3 | Win on draw 4 | |
The cases are disjoint, so add their probabilities.
Sum the infinite GP
. The bracket has first term 1, ratio 5/7 and infinitely many terms. Thus .
Common mistakes
The second red probability is 1/6, not 2/7. Red balls are removed even when the eventual consecutive-red condition fails.
Solution notes
Complete source table, first three winning cases/ellipsis, factored infinite-GP stages, parameters and printed mark 3 retained. Why nonconsecutive reds cannot win is made explicit from the replacement rules.