Worked solution
Game winnings, a mode and a tournament cutoff
A bag contains 4 red, 5 blue and 6 green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is played by performing 10 trials in all.
The player receives $0.50 for every blue ball drawn in a game. He receives nothing if balls of other colours are drawn.
(a) Find the probability that, in a game, he receives
(i) exactly $3,
(ii) more than $2.50,
(b) Find also the amount a player is most likely to receive.
(c) At the start of a tournament, each player plays the above game once. Players who earned more than $ proceed to the next round. Find the least value of such that, in a random sample of 10 players, the probability that all the 10 players proceed to the next round is less than 0.1.
CJC · Discussion · Question 8
Answer
(a)(i) ; (ii) . (b) $1.50. (c) (a $1 cutoff).
(a)(i)
Let X count blue draws in ten trials. Replacement gives . Exactly $3 means X=6, so (3 s.f.).
(a)(ii)
More than $2.50 means X>5. Thus (3 s.f.).
(b)
The printed mode table is:
| x | P(X=x) |
|---|---|
| 2 | 0.1951 |
| 3 | 0.2601 |
| 4 | 0.2276 |
The mode is X=3, so the most likely receipt is 3×$0.50=$1.50.
(c)
For independent players the required condition is . Equivalently . The source compares these cutoff values:
| k | 1−P(X≤2k) |
|---|---|
| 0.5 | 0.8960 |
| 1 | 0.7009 |
| 1.5 | 0.4407 |
At k=0.5, the ten-player probability is 0.333304913, which fails; at k=1 it is 0.0285959233, which passes. Since earnings come in half-dollar steps and progression requires strictly more than the cutoff, the event changes at these steps. Any k<1 admits at least X≥2 and fails, while k=1 requires X≥3. The least k is 1, corresponding to a $1 cutoff. A half-dollar increment in the calculator table is sufficient to find the event changes.
Common mistakes
“More than” makes the threshold strict. Threshold changes occur at attainable earnings, not at arbitrary integer counts of dollars.
Solution notes
The highlighted source note says “2k is an integer ⇒ k is not an integer.” This implication is false (k=1 is the source’s own answer). The useful instruction is to inspect half-dollar cutoff steps. No non-integer restriction on k is added. The answer k=1 is also the least real cutoff, because the passing event changes there.