The workings:
As it has become a tradition, the IMUS blog is taking a summer break. Whether it is deserved or not is up to our (dear) readers to decide. And as in previous years, in this particular section we are proposing a special divertimento, to carry on playing with mathematics as the stars of our problem. We hope that, despite the difficulties and uncertainties that surround us, we will all be able to find some rest and refreshment to come back to work in September.
The fun:
Anna and Beatrice are playing a heads or tails coin game, tossing alternately. When Anne has tossed for the \((n+1)\)-th time, they have to interrupt the game without Beatrice being able to continue. Anna then suggests the following deal: she has won if she has rolled more heads than Beatrice, with Beatrice being the winner if she has rolled at least the same amount as Anna. Is this a fair deal? (We assume that the coin is perfectly balanced.)
Speaking of fair games, they comment on a curiosity they read in the press: in the recent elections to the Galician Parliament, there was a district in which party A obtained \(a\) votes, party B obtained \(b\) votes and, moreover, in the vote count it was always the first party above the second. What was the probability of this happening?
.
.
.
.
.
.
Solutions
We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.
Tras conocer el voto $$t$$ (con $$t \in \{1, \dots, a+b\}$$) no me queda claro si el partido A tiene estrictamente mas votos que el partido B ó mayor o igual. Por ejemplo, si $$a = 2$$ y $$b = 1$$ la secuencia $$ABA$$ valdría o solo valdría la $$AAB$$. Y en tal caso supongo que $$a \geq b$$ en lugar de $$a > b$$. ¿Verdad?
Tras conocer el voto $t$ (con $t \in \{1, \dots, a+b\}$) no me queda claro si el partido A tiene estrictamente mas votos que el partido B ó mayor o igual. Por ejemplo, si $a = 2$ y $b = 1$ la secuencia $ABA$ valdría o solo valdría la $AAB$. Y en tal caso supongo que $a \geq b$ en lugar de $a > b$. ¿Verdad?
Vaya pues se va a quedar malamente escrito de igual forma.
Hola, Jaime, solo la secuencia AAB es válida, porque si el recuento sigue como ABA, al registrar el segundo voto A está empatado con B, no por encima.