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Chapter 7: Problem 24

Wieviel Tippreihen gibt es beim Fu脽balltoto (13-er Wette)? (Bei der 13-er Wette wird jedem der 13 Spiele, die auf dem Totoschein aufgef眉hrt sind, eine 0,1 oder 2 zugeordnet.)

Short Answer

Expert verified
There are 1,594,323 possible tip combinations.

Step by step solution

01

Understanding the Problem

We are asked to determine how many possible tip combinations are available for a 13-game football pool. Each game can result in one of three outcomes: home win (1), away win (2), or draw (0). This implies for each of the 13 games, there are 3 choices.
02

Formulating the Problem Mathematically

For each of the 13 games, there are 3 possible outcomes. Therefore, we need to calculate the total number of combinations by multiplying the probabilities for each game. Mathematically, this can be expressed as a power: the number of games (13) raised to the number of outcomes per game (3): \[ 3^{13} \].
03

Calculating the Total Combinations

Now let's compute the total number of possible combinations using the formula derived in the previous step. This involves calculating:\[ 3^{13} = 1,594,323 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Probability Theory is a branch of mathematics that deals with the likelihood of different outcomes occurring. In the context of sports betting, like the 13-game football pool highlighted in our problem, it helps us understand the range of possible outcomes. Each game can end in a home win, away win, or draw. Thus, the probability for each game is uniform, with each outcome having an equal chance of one-third.To determine the total number of combinations, we consider all potential outcomes' probabilities. Each game鈥檚 outcome is an independent event, meaning the result of one game does not affect the others. This allows us to calculate the total possibilities by multiplying the number of outcomes per game (3) to the power of the number of games: \[3^{13}\].This exponential function tells us that each game adds a new layer of outcomes, exponentially increasing the total number of possibilities. Hence, understanding probability enables us to recognize and calculate the extensive number of potential betting configurations in practical scenarios, such as the football pool.
Mathematical Problem Solving
Mathematical Problem Solving involves taking a problem and systematically working through the steps to find a solution. In the 13-game football pool problem, we start by identifying what is being asked: the number of possible combinations for game outcomes.First, we model the problem mathematically, understanding that each game can result in three possible outcomes. This transforms the problem into an exponential expression representing all potential combinations: \[3^{13}\].By structuring the problem step-by-step:
  • Identify the variables: Number of games and possible outcomes per game.
  • Formulate the expression: Use the exponential model to understand combination growth.
  • Perform the calculation: Compute the total combination count using the formula derived.
These steps illustrate a logical progression vital for solving mathematical problems, breaking them into manageable tasks, and finding the total number of outcomes effectively.
Sports Betting Mathematics
Sports Betting Mathematics applies mathematical principles to betting scenarios, helping to predict outcomes and formulate strategy. In the case of the 13-game football pool, mathematics helps us calculate all possible betting outcomes.Each game possibility (0, 1, 2) represents a basic unit of a sports bet. Consequently, understanding combinations and permutations becomes crucial in maximizing betting strategies. For example, the betting pool involves assigning one of these outcomes to each of the 13 games, recognized as a full permutation of the games with outcomes:\[3^{13}\].Understanding these betting mathematics aids bettors in determining the number of tipping lines available, illustrating just how many unique configurations exist. This knowledge helps in strategizing which combinations to bet on by analyzing potential returns and risks.In real-world applications, correctly applying sports betting algorithms and probability theory can provide a competitive edge, particularly in complex betting environments like multi-game pools.

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Most popular questions from this chapter

Sei \(x \neq y .\) Dann ist f眉r jedes nat眉rliche \(n>1\) $$ \frac{x^{n}-y^{n}}{x-y}=x^{n-1}+x^{n-2} y+x^{n-3} y^{2}+\cdots+x y^{n-2}+y^{n-1} $$ F眉r \(n=2\) ist dies nichts anderes als die wohlbekannte Gleichung \(x^{2}-y^{2}=(x+y)(x-y)\).

Beweise induktiv die nachstehende Verallgemeinerung der Bemoullischen Ungleichung: Sind die reellen Zahlen \(x_{1}, \ldots, x_{n}(n \geqslant 2)\) alle positiv oder alle negativ, aber \(>-1\), so ist $$ \left(1+x_{1}\right)\left(1+x_{2}\right) \cdots\left(1+x_{n}\right)>1+x_{1}+x_{2}+\cdots+x_{n} $$

Der Binomialkoeffizient \(\left(\begin{array}{l}\alpha \\\ k\end{array}\right)=\frac{\alpha(\alpha-1) \cdots(\alpha-k+1)}{1 \cdot 2 \cdots k}\) ist, locker formuliert, ein Bruch, dessen Z盲hler aus \(k\) absteigenden und dessen Nenner aus \(k\) aufsteigenden Faktoren besteht. Man lasse es sich nicht verdrie脽en, einige Binomialkoeffizienten auszurechnen (und dabei ausgiebig zu k眉rzen), z.B. $$ \left(\begin{array}{l} 5 \\ 3 \end{array}\right),\left(\begin{array}{c} 10 \\ 5 \end{array}\right),\left(\begin{array}{c} 1 / 2 \\ 2 \end{array}\right),\left(\begin{array}{c} -1 / 3 \\ 4 \end{array}\right),\left(\begin{array}{c} 4 \\ 5 \end{array}\right),\left(\begin{array}{c} -4 \\ 5 \end{array}\right),\left(\begin{array}{c} -2 / 3 \\ 3 \end{array}\right),\left(\begin{array}{c} 7 / 8 \\ 0 \end{array}\right) $$ Zeige: F眉r \(\alpha \neq 0\) ist \(\left(\begin{array}{c}-\alpha \\\ k\end{array}\right)=(-1)^{k}\left(\begin{array}{c}\alpha+k-1 \\\ k\end{array}\right)\)

20\. Auch wenn \(n\) gegebene Objekte \(a, b, \ldots\) nicht alle voneinander verschieden sind, nennt man jede Anordnung derselben (also jede Verteilung auf \(n\) K盲stchen \(\left.K_{1}, \quad, K_{n}\right)\) eine Permutation der \(a, b, \ldots\) Drei verschiedene Buchstaben \(a, b, c\) besitzen \(3 !=6\) Permutationen, die Buchstaben \(a, a, b\) jedoch nur noch drei: \(a a b, a b a, b a a\). Bestimme alle (verschiedenen) Permutationen der Buchstaben \(a, a, a, b\) und ebenso der Buchstaben \(a, a\), \(b, b .\) Z盲hle ab, wie viele es gibt und vgl mit Aufgabe \(21 .\)

Jedem Telefonanschlu脽 ist eine Vorwahlnummer und eine Telefonnummer zugeordnet. Wir nehmen vereinfachend an, die Vorwahlnummer bestehe aus f眉nf der , Ziffern" \(0,1, \ldots, 9\) mit der Ma脽gabe, da脽 die erste Ziffer stets \(=0\), die zweite stets \(\neq 0\) sein soll, und die Telefonnummer bestehe aus mindestens drei, aber h枚chstens f眉nf Ziffern, wobei die erste immer \(\neq 0\) sei Wieviel Telefonanschl眉sse sind grunds盲tzlich m枚glich?
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