Ahem!. Well, I know that there’s some brickbats in store for me today..:P. This is my second math misery this week [Lolz!]. But, yeah- all said; I still feel that this one is pretty lucid and perhaps a bit spicy too- Thus paving a pathway for a nice rapot!.
Coming to the theme, I always wondered as to why probability of success could not be calculated. Now, let me explain- Last week I installed some utterly crap symbian s60v2 game on to my N72, it had these so called real-life congruent-mirror-technology behind it; The aim of the game was to help a “person”[who happens to be the hero of the game!] achieve his expectations; If the explanation sounded drifting, I’ll sneak in an example scenario here, Its like if the “person” “expects” to win the Wimbledon grand slam, you need to make him achieve that!.
Initially I felt that the game had a lot of logical fallacies and some “pot holes” of horribly blatant blunders. But after the first round of novical play- I felt that this had a very high content of rich mathematics involved- Yes! Some really cool and interesting probabilistic approaches were on cards. So, as usual (jobless me)- I started musing about this, and to my amaze – the ideas started flowing at some extra-ordinary speed, my pipe-line was overflowing, yet the processor was stable and making some fabulous round of computations. And when it all ended after a mano-man of about 60minutes, I was done. And what I ended up formulating was the success probability of a person X in the Universe of Discourse. Here’s some insight into it---
Let's apply the notion of mathematical expectation to the example of a novice player seeking admittance to a tennis club. To be admitted, the fellow had to beat in two successive games members G (good) and T (top) of the club. With probabilities g and t (t < g) of winning against G and T, the fellow had to choose between to possible orders of games: GTG or TGT. Paradoxically, the second choice appeared to be preferable gaining the fellow the membership with the probability gt(2 - t) against the smaller gt(2 - g) for the sequence GTG.
We shall be looking for the expected number of wins. Using L for a loss and W for a win for the aspiring novice, we shall consider two sample spaces. Following the havils, the space consists of 8 possible outcomes of a sequence of three games:
LLL, LLW, LWL, LWW, WLL, WLW, WWL, WWW
However note that in the sequences LLL, LLW, WLL, WLW the third game is superfluous as the result of the first two make it impossible for the fellow to win two successive games, whereas the third game is unnecessary in the last two sequences WWL, WWW because the two first wins already gain the fellow admittance to the club. This makes possible and reasonable to consider a smaller sample space:
LL, LWL, LWW, WL, WW
For the sequence TGT we have the following probabilities:
Win/Loss sequence Probability
LLL (1 - t)(1 - g)(1 - t)
LLW (1 - t)(1 - g)t
LWL (1 - t)g(1 - t)
LWW (1 - t)gt
WLL t(1 - g)(1 - t)
WLW t(1 - g)t
WWL tg(1 - t)
WWW tgt
for the first sample space and
Win/Loss sequence Probability
LL (1 - t)(1 - g)
LWL (1 - t)g(1 - t)
LWW (1 - t)gt
WL t(1 - g)
WW tg
for the second. In both cases, the probabilities add up to 1, as required. Choosing the easier way out, we verify this only for the latter:
(1 - t)(1 - g) + (1 - t)g(1 - t) + (1 - t)gt + t(1 - g) + tg
= (1 - t)(1 - g) + [(1 - t)g(1 - t) + (1 - t)gt] + t(1 - g) + tg
= (1 - t)(1 - g) + (1 - t)g + t(1 - g) + tg
= [(1 - t)(1 - g) + t(1 - g)] + [(1 - t)g + tg]
= (1 - g) + g
= 1.
Now we introduce the random variable N that denotes the number of wins for the candidate. In the first case, N may be 0, 1, 2, or 3; in the second case, the are only three possible values: 0, 1,2.
The expectations E1 and E2 are
E1(N, TGT) = 0•(1 - t)(1 - g)(1 - t)
+ 1•(1 - t)(1 - g)t
+ 1•(1 - t)g(1 - t)
+ 2•(1 - t)gt
+ 1•t(1 - g)(1 - t)
+ 2•t(1 - g)t
+ 2•tg(1 - t)
+ 3•tgt
= 2t + g
and, correspondingly,
E2(N, TGT) = 0•(1 - t)(1 - g)
+ 1•(1 - t)g(1 - t)
+ 2•(1 - t)gt
+ 1•t(1 - g)
+ 2•tg
= t + g + tg - t2g.
Similarly,
E1(N, GTG) = t + 2g and
E2(N, GTG) = t + g + tg - tg2.
Since t < g, we see that
E1(N, TGT) < E1(N, GTG),
as expected (pun intended). We also have
E2(N, TGT) < E2(N, GTG),
which ameliorates the paradoxical situation that arose from the pure count of probabilities. Although, the probability of gaining the membership playing the top guy first is larger then when playing first just a good member, the expected number of the wins is greater when postponing the confrontation with the top player.
So whats astounding is that the player can actually have very high expectations of wining the tourney. Wow!, I was stunned!!- I have heard people bog down since they are under-dogs and few other yombering around saying that they have no chance what so ever of achieving their goals. Now, all you people out there, read this mathematical snippet and get inspired to conquer the world!. Your chances are not meager by any means, its just that you need to apply things and have a solidiifed state of mind!. Lolz and talking about solidification, I hope to follow this up with some proof for the same along Chemistry of Human Physical Biology..:P
But, till then- Hold your breath and say Sigh!- I can also beat Federer!(Ok, girls and more pontificially ladies you can use Ana Ivanovic too! :P)
