Question

A football team is known to run \(30 \%\) of its plays to the left and \(70 \%\) to the right. A linebacker on an opposing team notices that, when plays go to the right, the right guard shifts his stance most of the time \((80 \%)\) 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?

Short answer

Answer: The linebacker should prepare to defend the left side, as there is approximately a 65.9% chance the play will go to the left when the guard takes a balanced stance.

Step-by-step solution

Organize the given information

We are given the following probabilities: - The football team runs 30% of its plays to the left (P(Left) = 0.3) and 70% of its plays to the right (P(Right) = 0.7). - When plays go to the right, the right guard: - Takes a shift stance 80% of the time (P(Shift | Right) = 0.8) - Takes a balanced stance 20% of the time (P(Balanced | Right) = 0.2) - When plays go to the left, the right guard: - Takes a balanced stance 90% of the time (P(Balanced | Left) = 0.9) - Takes a shift stance 10% of the time (P(Shift | Left) = 0.1) Now, we need to find: a. The probability that the play will go to the left, given that the guard takes a balanced stance (P(Left | Balanced))) b. The probability that the play will go to the right, given that the guard takes a balanced stance (P(Right | Balanced))

Bayes' Theorem formula

Bayes' theorem relates the conditional probability of an event given another event. The formula is: $$ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} $$ We will apply this formula to our problem: a. P(Left | Balanced) = \(\frac{P(Balanced | Left) \cdot P(Left)}{P(Balanced)}\) b. P(Right | Balanced) = \(\frac{P(Balanced | Right) \cdot P(Right)}{P(Balanced)}\)

Calculate the probability of a balanced stance

We need to find P(Balanced), which can be calculated using the law of total probability: $$ P(Balanced) = P(Balanced | Left) \cdot P(Left) + P(Balanced | Right) \cdot P(Right) $$ $$ P(Balanced) = (0.9 \times 0.3) + (0.2 \times 0.7) = 0.27 + 0.14 = 0.41 $$

Calculate P(Left | Balanced) and P(Right | Balanced)

Now we can calculate the conditional probabilities: a. P(Left | Balanced) = \(\frac{P(Balanced | Left) \cdot P(Left)}{P(Balanced)} = \frac{0.9 \times 0.3}{0.41} \approx 0.659\) b. P(Right | Balanced) = \(\frac{P(Balanced | Right) \cdot P(Right)}{P(Balanced)} = \frac{0.2 \times 0.7}{0.41} \approx 0.341\)

Choose the direction to defend

Based on the calculated probabilities, if the linebacker saw the balanced stance, they should prepare to defend: - The left side, with a probability of approximately 65.9% - The right side, with a probability of approximately 34.1% Since the probability of the play going to the left is higher when the guard takes a balanced stance, the linebacker should prepare to defend the left side.

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that deals with the likelihood of an event occurring, given that another event has already occurred. This concept is crucial for situations where the outcome depends on a prior condition or event. In the football exercise, we determine the probability of which direction the play is going based on observing a balanced stance of the guard.

The formula for conditional probability looks like this: \( P(A | B) = \frac{P(A \cap B)}{P(B)} \), where \(P(A | B)\) is the probability of event A occurring given that event B has occurred.

• If we calculate the probability that the play goes to the left when we observe a balanced stance, we are essentially calculating \(P(\text{Left} | \text{Balanced})\).
• Using the probabilities given in the problem, such as \(P(\text{Balanced} | \text{Left}) = 0.9\) and \(P(\text{Left}) = 0.3\), we can find the conditional probability of the play going left.

Conditional probability requires understanding how different events relate to one another, helping us make predictions and informed decisions under uncertainty.

Law of Total Probability

The Law of Total Probability is a crucial principle that facilitates the calculation of probabilities for complex events. This law helps us determine the probability of a certain event by considering all possible scenarios or paths that could lead to that event. It essentially breaks down probabilities across different mutually exclusive events.

In the context of the exercise, we use this law to calculate \(P(\text{Balanced})\), the probability that the guard takes a balanced stance regardless of the play's direction.

The formula for the Law of Total Probability is: \[ P(B) = P(B | A_1) \cdot P(A_1) + P(B | A_2) \cdot P(A_2) + \ldots + P(B | A_n) \cdot P(A_n) \] The application involves:

• Considering each direction the play might go (left or right) as separate scenarios, each with its own probability.
• Calculating the combined probability that results in a balanced stance across these different scenarios.

This principle is powerful because it transforms complicated probability questions into manageable parts, giving us a fuller picture of the event gathering points from different conditions.

Probability Theory

Probability theory is the branch of mathematics that studies the likelihood of different outcomes in uncertain scenarios. It's the foundation of concepts such as conditional probability and the Law of Total Probability, which are used to analyze the football game scenario.

In probability theory, we measure how likely an event is to occur. This is done using:

• Probabilities that range from 0 to 1, where 0 means an event is impossible, and 1 means it is certain.
• Tools and formulas like Bayes' Theorem are employed to update our beliefs based on new information.
• Understanding chance and random events underpins decision making in varied fields like gambling, weather forecasting, finance, and science.

For the exercise in question, probability theory allows us to determine the most probable direction of the play based on previous observations (the guard's stance). This means we are not just guessing, but using structured methods to make informed predictions about possible outcomes.