Here are 64 teams who play single elimination tournaments, hence6 rounds, and you have to predict all the winners in all 63 games. your scoreis then computed as follows: 32 points for correctly predicting the final winner,16 points for each correct finalist, and so on, down to 1 point of every correctlypredicted winner for the first round. (the maximum number of points you canget is thus 192.) knowing nothing about any team, you flip fair coins to decideevery one of your 63 bets. compute the expected number of points.

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chrisayy

19.02.2023

Mathematics

$\bf \begin{cases}
\cfrac{2}{3}x+y=6\\[0.8em]
-\cfrac{2}{3}x-y=2
\end{cases}
\\\\[-0.35em]
\rule{34em}{0.25pt}\\\\
\stackrel{\textit{using the first equation}}{\cfrac{2}{3}x+y=6}\implies y=6-\cfrac{2x}{3}
\\\\[-0.35em]
~\dotfill$

$\bf \stackrel{\textit{using the found \underline{y} in the second equation}}{-\cfrac{2x}{3}-\left( 6-\cfrac{2x}{3} \right)=2}\implies -\cfrac{2x}{3}-6+\cfrac{2x}{3}=2
\\\\\\
\stackrel{\textit{multiplying both sides by }\stackrel{LCD}{3}}{-2x-18+2x=6}\implies -18\ne 6\impliedby \stackrel{\textit{system is inconsistent}}{\textit{no solutions}}$

if you set both equations to a y = mx+b, you'll notice the slope is the same, meaning the lines are parallel to each other, thus they never meet.

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