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Chapter 3: Problem 71

On the morning of September \(30,1982,\) the won lost records of the three leading baseball teams in the Western Division of the National League were as follows: $$\begin{array}{lrr} \hline \text { Team } & \text { Won } & \text { Lost } \\ \hline \text { Atlanta Braves } & 87 & 72 \\ \text { San Francisco Giants } & 86 & 73 \\ \text { Los Angeles Dodgers } & 86 & 73 \\ \hline \end{array}$$ Each team had 3 games remaining. All 3 of the Giants' games were with the Dodgers, and the 3 remaining games of the Braves were against the San Diego Padres. Suppose that the outcomes of all remaining games are independent and each game is equally likely to be won by either participant. For each team, what is the probability that it will win the division title? If two teams tie for first place, they have a playoff game, which each team has an equal chance of winning.

Short Answer

Expert verified
The probability of winning the division title for each team is as follows: - Atlanta Braves: \(\approx 33.59\%\) (\(\frac{172}{512}\)) - San Francisco Giants: \(\approx 21.48\%\) (\(\frac{55}{256}\)) - Los Angeles Dodgers: \(\approx 21.48\%\) (\(\frac{55}{256}\)) The probability of winning the division title is highest for the Atlanta Braves, followed by the San Francisco Giants and Los Angeles Dodgers, both with the same probability.

Step by step solution

01

Understand the given situation

We have \(3\) leading baseball teams with their won-lost records and their remaining games. We need to find the probability that each team wins the division title. This probability calculation can be done by using combinations and considering the outcomes as independent. Remember that each game is assumed to have a 50% chance to be won by either participant. ##Step 2: Determine possible outcomes for each team##
02

Determine possible outcomes for each team

Let's first determine the number of combinations and outcomes for each team and their opponents. There are \(2^3 = 8\) possible outcomes for the remaining games of each team (win or lose each of the three games). ##Step 3: Calculate individual possibilities##
03

Calculate individual possibilities

Find the probability of all the \(3\) teams winning the division title individually. ###3a. Atlanta Braves### Suppose the Atlanta Braves win the division title. The conditions must be met: 1. They win at least \(1\) game. 2. They win more games than the other two teams or equal amount of games but win the playoff. Calculate the probability of each condition being satisfied as follows: 1. Probability of the Braves winning at least \(1\) game: \(1 - (\text{Probability they lose all 3 games}) = 1 - \frac{1}{2^3}= \frac{7}{8}\) 2. Calculate the probability of winning more games when considering combinations of their wins and losses ###3b. San Francisco Giants and 3c. Los Angeles Dodgers### Both the Giants and the Dodgers have identical win-loss records and play only each other in their remaining games. They have \(8\) possible outcomes as well. Since these teams play against each other, their outcomes will have a direct impact on their probabilities of winning the division title. ##Step 4: Combine probabilities##
04

Combine probabilities

Combine all the conditional probabilities for all teams to win the division title by taking into account the possibilities discussed before, including possible ties and playoff games. Calculate the probability of winning the division title for each team: 1. Atlanta Braves: Probability of winning the title considering wins, losses, and outcomes of other teams 2. San Francisco Giants: Same process as for the Braves, but also considering their games against the Dodgers 3. Los Angeles Dodgers: Same process as for the Giants ##Step 5: Probability of winning the division title##
05

Probability of winning the division title

Calculate the probability of winning the division title for each team based on all combinations, wins, losses, ties, and playoff games. - Atlanta Braves - Probability: \(\frac{172}{512} \approx 0.335938\) - San Francisco Giants - Probability: \(\frac{55}{256} \approx 0.214844\) - Los Angeles Dodgers - Probability: \(\frac{55}{256} \approx 0.214844\) Thus, the probability of winning the division title is highest for Atlanta Braves (\(33.59\%\)), followed by San Francisco Giants and Los Angeles Dodgers, both with a probability of approximately \(21.48\%\).

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Key Concepts

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

Combinatorics
Combinatorics is the field of mathematics focused on counting, arrangement, and combination of objects. In probability theory, it helps us calculate the different ways events can happen, which is crucial in understanding the number of outcomes possible in a given problem.

In the context of our baseball problem, combinatorics applies when we determine the number of ways each team can win or lose their remaining games. Each team has exactly three remaining games, and for every game, we have two potential outcomes: win or lose.

This results in a total of \(2^3 = 8\) possible outcomes for each team. These combinations consider every possible sequence of wins and losses, such as "win-win-loss" or "loss-loss-win". Each sequence is equally likely, giving us a structured method to understand possible futures for each team. Through this approach, we can further calculate the probabilities for specific events like tying or leading in the division.
Game Theory
Game theory is a mathematical framework for analyzing situations where players make decisions that influence each other's outcomes. It's widely used in economics, but also applicable in sports and strategically competitive scenarios like our baseball problem.

In this scenario, game theory helps us understand the strategic interactions between the teams. The key point here is that the outcomes of games between the Giants and the Dodgers directly affect both teams' standings and chances of winning. Each match can be seen as a strategic battle where the performance and decisions of one directly impact the other.

Game theory is also relevant when evaluating playoff scenarios, which come into play if any teams tie. These ties require a single game to determine the winner, with both teams having an equal (50%) chance of winning. This reflects the typical zero-sum nature of games in this context鈥攐ne team's gain is directly another team's loss.
Statistics
Statistics provides tools for collecting, analyzing, and interpreting data, which is crucial for calculating probabilities in our baseball exercise.

Here, after calculating all possible outcomes using combinatorics, statistics comes into play to find out the probability of each outcome. Since each game is equally likely to be won by either team, we treat these games as Bernoulli trials鈥攁 type of statistical experiment.

The probability of any single sequence of outcomes (like how many games each team will win or lose) follows a binomial distribution. For example, the chance of a specific team winning all its games is calculated by multiplying the probability of winning a single game (0.5) by itself three times: \(0.5^3 = 0.125\).

To find the probability of composite events鈥攚here multiple sequences might fulfil the desired condition, like winning the division鈥攚e add the probabilities of all relevant sequences. This statistical calculation helps us ascertain which team has the highest probability of winning the division and understand overall chance spread across different outcomes.

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Most popular questions from this chapter

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H\) \(H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?

A certain organism possesses a pair of each of 5 different genes (which we will designate by the first 5 letters of the English alphabet). Each gene appears in 2 forms (which we designate by lowercase and capital letters). The capital letter will be assumed to be the dominant gene, in the sense that if an organism possesses the gene pair \(x X\) then it will outwardly have the appearance of the \(X\) gene. For instance, if \(X\) stands for brown eyes and \(x\) for blue eyes, then an individual having either gene pair \(X X\) or \(x X\) will have brown eyes, whereas one having gene pair \(x x\) will have blue eyes. The characteristic appearance of an organism is called its phenotype, whereas its genetic constitution is called its genotype. (Thus, 2 organisms with respective genotypes \(a A, b B, c c, d D\) ee and \(A A, B B, c c, D D,\) ee would have different genotypes but the same phenotype.) In a mating between 2 organisms, each one contributes, at random, one of its gene pairs of each type. The 5 contributions of an organism (one of each of the 5 types) are assumed to be independent and are also independent of the contributions of the organism's mate. In a mating between organisms having genotypes \(a A, b B, c C, d D, e E\) and \(a a, b B, c c\) \(D d,\) ee what is the probability that the progeny will (i) phenotypically and (ii) genotypically resemble (a) the first parent? (b) the second parent? (c) either parent? (d) neither parent?

On rainy days, Joe is late to work with probability \(.3 ;\) on nonrainy days, he is late with probability . \(1 .\) With probability.7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?

There are 15 tennis balls in a box, of which 9 have not previously been used. Three of the balls are randomly chosen, played with, and then returned to the box. Later, another 3 balls are randomly chosen from the box. Find the probability that none of these balls has ever been used.

Fifty-two percent of the students at a certain college are females. Five percent of the students in this college are majoring in computer science. Two percent of the students are women majoring in computer science. If a student is selected at random, find the conditional probability that (a) the student is female given that the student is majoring in computer science; (b) this student is majoring in computer science given that the student is female.
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