|
Call the probability of Justine's making her point p (equal to 0.80 in this case), and that her opponent will take the point as q (0.20 in this case).
Call the probability of Justine's winning the game from deuce d. And from ad iN (advantage Justine), n, and from ad ouT, t.
The probability is p that deuce will be converted to ad in, and q that it will be converted to ad out. Likewise the probability is p that ad in would be converted to a win, and q that it would be converted to deuce. Finally it is probability p that ad out would be converted to deuce and q that it would be converted to a loss. So:
d = pn + qt
n = p + qd
t = pd
so
d = p(p+qd) + qpd
d = p^2 + pqd + pqd
d - 2pqd = p^2
d(1-2pq) = p^2
d = p^2 / (1-2pq)
Once you have d, you can find n and t from the remaining formulae above.
In the case of p = 0.80, or 8/10, the probability of a win from deuce comes out to 16/17 or about .941176; from ad in, 84/85 or about .988235; and from ad out, 64/85 or about .752941.
Probabilities from lower given scores can be found by proceding backwards. For example, p(40-15) = p + q*p(40-30) = p + qd. (The hyphens indicate the separator of the scores, not subtraction). Or p(30-15) = p*p(40-15) + q*p(30-30). (Where p(30-30) is of course p*p(40-30) + q*p(30-40).)
Going backward in this manner, the probability, calculated at the beginning of the game, that Justine wins is 0.978221.
A set of tables of probabilities follows. Each table is headed by the percent of points that are expected to be scored by the server, and shows the probability of the server's winning the game, given any state of the game (score thus far) that might occur in the game.
p 50% 60% 70% 80% 90%
from
0- 0 0.500000 0- 0 0.735729 0- 0 0.900789 0- 0 0.978221 0- 0 0.998552
0-15 0.343750 0-15 0.576222 0-15 0.789018 0-15 0.930033 0-15 0.990551
0-30 0.187500 0-30 0.368862 0-30 0.587831 0-30 0.795106 0-30 0.944144
0-40 0.062500 0-40 0.149538 0-40 0.289776 0-40 0.481882 0-40 0.720110
15- 0 0.656250 15- 0 0.842068 15- 0 0.948691 15- 0 0.990268 15- 0 0.999441
15-15 0.500000 15-15 0.714462 15-15 0.875241 15-15 0.963765 15-15 0.995707
15-30 0.312500 15-30 0.515077 15-30 0.715569 15-30 0.873412 15-30 0.969037
15-40 0.125000 15-40 0.249231 15-40 0.413966 15-40 0.602353 15-40 0.800122
30- 0 0.812500 30- 0 0.927138 30- 0 0.980169 30- 0 0.996894 30- 0 0.999856
30-15 0.687500 30-15 0.847385 30-15 0.943672 30-15 0.986353 30-15 0.998671
30-30 0.500000 30-30 0.692308 30-30 0.844828 30-30 0.941176 30-30 0.987805
30-40 0.250000 30-40 0.415385 30-40 0.591379 30-40 0.752941 30-40 0.889024
40- 0 0.937500 40- 0 0.980308 40- 0 0.995810 40- 0 0.999529 40- 0 0.999988
40-15 0.875000 40-15 0.950769 40-15 0.986034 40-15 0.997647 40-15 0.999878
40-30 0.750000 40-30 0.876923 40-30 0.953448 40-30 0.988235 40-30 0.998780
40-40 0.500000 40-40 0.692308 40-40 0.844828 40-40 0.941176 40-40 0.987805
p 55% 65% 75% 85% 95%
from
0- 0 0.623149 0- 0 0.829645 0- 0 0.949219 0- 0 0.992863 0- 0 0.999908
0-15 0.458030 0-15 0.689214 0-15 0.870117 0-15 0.969215 0-15 0.998783
0-30 0.270895 0-30 0.476563 0-30 0.696094 0-30 0.879353 0-30 0.985508
0-40 0.099660 0-40 0.212897 0-40 0.379688 0-40 0.595578 0-40 0.855007
15- 0 0.758246 15- 0 0.905261 15- 0 0.975586 15- 0 0.997036 15- 0 0.999967
15-15 0.611140 15-15 0.803719 15-15 0.928125 15-15 0.985073 15-15 0.999481
15-30 0.410996 15-30 0.618536 15-30 0.801563 15-30 0.929431 15-30 0.992376
15-40 0.181200 15-40 0.327534 15-40 0.506250 15-40 0.700680 15-40 0.900007
30- 0 0.878605 30- 0 0.959937 30- 0 0.991406 30- 0 0.999147 30- 0 0.999992
30-15 0.774894 30-15 0.903433 30-15 0.970313 30-15 0.994892 30-15 0.999855
30-30 0.599010 30-30 0.775229 30-30 0.900000 30-30 0.969799 30-30 0.997238
30-40 0.329455 30-40 0.503899 30-40 0.675000 30-40 0.824329 30-40 0.947376
40- 0 0.963460 40- 0 0.990363 40- 0 0.998438 40- 0 0.999898 40- 0 1.000000
40-15 0.918800 40-15 0.972466 40-15 0.993750 40-15 0.999320 40-15 0.999993
40-30 0.819554 40-30 0.921330 40-30 0.975000 40-30 0.995470 40-30 0.999862
40-40 0.599010 40-40 0.775229 40-40 0.900000 40-40 0.969799 40-40 0.997238
So for Maria serving Justine, Maria's chance of winning the game from deuce would be 90%; from ad in or 40-30, 97.5% (or 39/40) and from ad out or 30-40, 67.5% (or 27/40). At the beginning of the game, you'd say 94.9219...%.
In rational numbers, the relevant figures are:
80 %
p = 4/5; q = 1/5
from prob of win
40-40 (deuce) 16/17
40-30 84/85
40-15 p + q*p(40-30) 424/425
40-0 p + q*p(40-15) 2124/2125
30-40 64/85
30-30 p*p(40-30) + q*p(30-40) 16/17
30-15 p*p(40-15) + q*p(30-30) 2096/2125
30-0 p*p(40-0) + q*p(30-15) 10592/10625
15-40 p*p(30-40) 256/425
15-30 p*p(30-30) + q*p(15-40) 1856/2125
15-15 p*p(30-15) + q*p(15-30) 2048/2125
15-0 p*p(30-0) + q*p(15-15) 52608/53125
0-40 p*p(15-40) 1024/2125
0-30 p*p(15-30) + q*p(0-40) 8448/10625
0-15 p*p(15-15) + q*p(0-30) 49408/53125
0-0 p*p(15-0) + q*p(0-15) 51968/53125
75 %
p = 3/4; q = 1/4
from prob of win
40-40 (deuce) 9/10
40-30 39/40
40-15 p + q*p(40-30) 159/160
40-0 p + q*p(40-15) 639/640
30-40 27/40
30-30 p*p(40-30) + q*p(30-40) 9/10
30-15 p*p(40-15) + q*p(30-30) 621/640
30-0 p*p(40-0) + q*p(30-15) 1269/1280
15-40 p*p(30-40) 81/160
15-30 p*p(30-30) + q*p(15-40) 513/640
15-15 p*p(30-15) + q*p(15-30) 297/320
15-0 p*p(30-0) + q*p(15-15) 999/1024
0-40 p*p(15-40) 243/640
0-30 p*p(15-30) + q*p(0-40) 891/1280
0-15 p*p(15-15) + q*p(0-30) 891/1024
0-0 p*p(15-0) + q*p(0-15) 243/256
This puzzle is not based on any actual statistics of any real people named Justine or Maria. It was inspired by a supposedly counterintuitive result in Julian Havil's Book, Nonplussed: Mathematical Proof of Implausible Ideas, that when a player has a probability of winning the point on her service greater than 0.919643... (p > 0.919643..., the root of the equation 8p^3 - 4p^2 - 2p - 1 = 0), then p(0-0) > p(40-30). But when p > 0.919643, the value of p(0-0) is over .999391, which means that her service would be broken only once in at least 1642 games; the players are clearly not well matched, and it would be amazing if it got to 40-30; it would be hard enough for the lesser player to get to merely 40-15. Mr. Havil appears to believe that such high percentages would apply among evenly matched players, but that's not the case: the server advantage is not as great as he thinks it is.
DECLARE FUNCTION pWin# (a#, b#)
DEFDBL A-Z
CLEAR , , 9999
DIM score(4)
DIM SHARED p, q, d, t, n
score(1) = 15
score(2) = 30
score(3) = 40
score(4) = 99
CLS
FOR pp = 50 TO 90 STEP 10
p = pp / 100
q = 1 - p
d = p * p / (1 - 2 * p * q)
t = d * p
n = p + q * d
col = (pp / 10 - 5) * 15 + 1
LOCATE 2, col + 6: PRINT USING "##&"; pp; "%";
ln = 2
FOR own = 0 TO 3
FOR opp = 0 TO 3
LOCATE 2 + ln, col
PRINT USING "##&## #.######"; score(own); "-"; score(opp); pWin(own, opp)
ln = ln + 1
NEXT
NEXT
NEXT
FOR pp = 55 TO 95 STEP 10
p = pp / 100
q = 1 - p
d = p * p / (1 - 2 * p * q)
t = d * p
n = p + q * d
col = (pp / 10 - 5.5) * 15 + 1
LOCATE 21, col + 6: PRINT USING "##&"; pp; "%";
ln = 21
FOR own = 0 TO 3
FOR opp = 0 TO 3
LOCATE 2 + ln, col
PRINT USING "##&## #.######"; score(own); "-"; score(opp); pWin(own, opp)
ln = ln + 1
NEXT
NEXT
NEXT
FUNCTION pWin (a, b)
IF a = 3 AND b = 3 THEN
pWin = d
ELSEIF a = 3 AND b = 2 THEN
pWin = n
ELSEIF a = 2 AND b = 3 THEN
pWin = t
ELSEIF a = 4 THEN
pWin = 1
ELSEIF b = 4 THEN
pWin = 0
ELSE
pWin = p * pWin(a + 1, b) + q * pWin(a, b + 1)
END IF
END FUNCTION
|
