91Ó°ÊÓ

A particular football team is known to run \(30 \%\) of its plays to the left and \(70 \%\) to the right. A linebacker on an opposing team notes that the right guard shifts his stance most of the time \((80 \%)\) when plays go to the right and that he uses a balanced stance the remainder of the time. When plays go to the left, the guard takes a balanced stance \(90 \%\) of the time and the shift stance the remaining \(10 \%\). On a particular play, the linebacker notes that the guard takes a balanced stance. a. What is the probability that the play will go to the left? b. What is the probability that the play will go to the right? c. If you were the linebacker, which direction would you prepare to defend if you saw the balanced stance?

Step by step solution

01

Define events and given probabilities

Let's define two events: - L: the play will go to the left - R: the play will go to the right The given probabilities are: - P(L) = 0.3 - P(R) = 0.7 - P(shift | R) = 0.8, this means that P(balanced | R) = 0.2 - P(balanced | L) = 0.9 - P(shift | L) = 0.1

02

Apply Bayes' theorem for P(L | balanced)

We're asked to find P(L | balanced), so we will apply Bayes' theorem: P(L | balanced) = P(balanced | L) * P(L) / P(balanced)

03

Calculate P(balanced)

To find P(balanced), we first need to find it using the total probability theorem: P(balanced) = P(balanced | L) * P(L) + P(balanced | R) * P(R) Plugging in the given values: P(balanced) = 0.9 * 0.3 + 0.2 * 0.7 = 0.47

04

Calculate P(L | balanced)

Now we can calculate P(L | balanced) using the given values and P(balanced): P(L | balanced) = (0.9 * 0.3) / 0.47 ≈ 0.574 So, the probability that the play will go to the left when the guard takes a balanced stance is approximately 0.574.

05

Apply Bayes' theorem for P(R | balanced)

Similarly, we're asked to find P(R | balanced), so we will apply Bayes' theorem again: P(R | balanced) = P(balanced | R) * P(R) / P(balanced)

06

Calculate P(R | balanced)

Now we can calculate P(R | balanced) using the given values and P(balanced): P(R | balanced) = (0.2 * 0.7) / 0.47 ≈ 0.426 So, the probability that the play will go to the right when the guard takes a balanced stance is approximately 0.426.

07

Determine which direction to prepare to defend

If you were the linebacker and saw the balanced stance, you should prepare to defend the direction with the higher probability, which is the left in this case: P(L | balanced) ≈ 0.574 > P(R | balanced) ≈ 0.426 So, the linebacker should prepare to defend to the left.

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Key Concepts

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

Conditional Probability

Conditional probability is a fundamental concept in probability and statistics that involves calculating the probability of an event, given that another event has occurred. This is represented as P(A|B), the probability of event A occurring given that event B is true.

For instance, in the football scenario, we examine the probability that the play will go to the left (L) given that the guard takes a balanced stance. It's important because it allows us to update our predictions based on new information, which is exactly what the linebacker does. By observing the guard's position, the linebacker gains insight into the team's likely direction of play.

Total Probability Theorem

The total probability theorem is a powerful tool in probability theory that helps calculate the overall likelihood of an event based on several different, mutually exclusive scenarios. It's expressed as the sum of the probabilities of the event occurring under each scenario, weighted by the probability of each scenario occurring.

In our example with the football team, we needed to consider two scenarios: play going to the left and play going to the right. By combining the guard's probabilities of his stance in both scenarios, we can determine the total likelihood of him taking a balanced stance. This comprehensive approach is essential for analyzing complex systems where outcomes can be influenced by multiple factors.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with analyzing random events and data. While probability focuses on predicting the likelihood of future events, statistics is about analyzing past data. These disciplines are intertwined; for instance, conditional probability and the total probability theorem are key concepts within both fields.

Understanding these principles is essential for interpreting data and making decisions based on incomplete information, as we often do in everyday life and in fields such as science, economics, and engineering. The precise calculation of probabilities in the football team problem exemplifies how these mathematical tools guide reasoning in practical situations.

Decision Making in Probability

In the context of probability, decision-making often involves choosing the most likely outcome or the option that maximizes expected benefits. To do this, one uses the calculated probabilities to inform their choices, assessing risks and potential rewards.

For the linebacker in our exercise, decision making comes down to positioning themselves to intercept the play's most probable direction based on the guard's stance. This process illustrates how probability aids in making informed strategic choices, a practice applied across various disciplines from finance to sports strategy, and emphasizes the practical significance of understanding probability concepts.