當我們這裡說"專業玩家 "即是表示他每投十套鏢平均最少總有一次拿到180分, 如圖所示,如果他們瞄準60位為目標, 他們所投的鏢與60位是非常接近, 最遠亦只是在高低一 二分之間. 換一句說話, 他們在投鏢時我們應只有鼓掌可以做. 世界上這個水平的真正專業高手不多.

高手所投的鏢仍然比較集中, 當他們瞄準20三倍時, 縱然不中目標, 亦不會超出1和5位以外. 而用平均機會率計算, 每100套鏢應可以得到一次180分. 平均分數74.7以上. 為了這種功夫他們應該花了很多時間在酒吧?

雖然普通玩家投鏢每套之間很接近, 但鏢與鏢之間的距離相差較大, 他們瞄準20三倍時,大致有十分之一可以射中目標.(不是指180分/套)

你是否上述似曾相識, 與你平日投鏢很接近?

雖然我們說初學者投中三倍分(treble)的機會率是4.64%, 但初學者 這組別亦是技術水平最參差不齊.

After solving how the darts are spread around the aiming point you must decide which kind of players you want to take into your calculations. I decided to use four different levels of players. Just to make it easier for you to know what level I am talking about, I have named these levels: professional, expert, average and beginner.

Since the dispersions of darts should be exact for finding the exact percentages for these levels, there must be something that makes them exact. I have chosen the probability of hitting the treble-bed. The 'professional's' probability of hitting the treble is hereby always 46.42% in the book. An expert can hit the treble with the probability of 21.54%. For an average player the same figure is 10.00% and for the beginner only 4.64%.

I have not chosen these numbers randomly: the percentage of a professional raised to the third power makes exactly 10%. You can thus say that a 'pro' can get 180 points - three darts to treble-bed - every tenth time he tries. As you may already have guessed an expert can score 180p every 100th time and so on.

                   probability of    probability of
                     hitting the       scoring 180
                       treble            points
------------------------------------------------------
professional (pro)     46.42%           1 to 10
expert (exp)           21.54%           1 to 100
average (ave)          10.00%           1 to 1000
beginner (beg)          4.64%           1 to 10000
------------------------------------------------------

You must, however, notice that the probability of getting 180 points is only theoretical: (46.4%^3 = 10%). After the first two darts there is already considerably less space on the treble-bed for the third dart. Therefore you often have to change your aiming point. Thus the 'right' probability of scoring the maximum score may be one-to-twenty or even less.

Then there is the mental pressure, which may be enormous. If you have never reached 180 points before, your hands may start to tremble. If you have practiced darts for many years already and have never scored 180, it is probably the biggest reason for this. It doesn't mean that you must now rank yourself in the 'beginners' group. In fact, even if you try to test the level of your play to find it out, there is no reason to start calling yourself 'an expert' or 'an average player'. As mentioned earlier, the names are used only because it's much easier to understand what kind of sprays I'm talking about. However, the test can help you to see what level you should be interested in.

On the other hand, after the first dart the beginner e.g. might have a good 'sighter' for his next shots, and the probability of throwing the remaining darts into the target can be much bigger than before. So, it's very hard to tell the exact level of scoring 180 points.

It would be more correct to say that an expert can hit three different trebles, for example T20, T15 and T16, every hundredth time. Since the theoretical probability is approximately right, it's easier to say that an expert can get 180 points every tenth time.

Probabilities

As you have noticed, everything in darts has something to do with probabilities. When you choose the way of finishing you just try to find the best probability to win the game. The probability of the opponent finishing the next turn is also important. You can estimate these probabilities in your head or you can even calculate them mathematically, as you have seen done in the book first time ever.

At first you must find out the dispersions of darts for the player. You can do that with a test, but that would require a lot of throws to be reliable. Or you can do it purely mathematically when you know the theory of normal dispersion. Since we want to make the results of optimal play as general as possible, we must use a circle as the form of the dispersion. One way or another you have to know what the player's probability of hitting for example a double is, how many darts then fly to a single, which percentage to the sector beside and so on. These probabilities have to be known for every aiming point you want to calculate.

When you have found a suitable spray of darts, you simply move it around the dart-board. When you place exactly the same dispersion to the centre of a double you get the percentage of hitting the double. Note that for the computer there are three aiming points in every bed of treble, double and single.

Naturally, you must also use the same spray of darts for the bull's-eye.