Question

Sports and weather. The San Francisco football team plays better in fair weather. They have a \(70 \%\) chance of winning in good weather, but only a \(20 \%\) chance of winning in bad weather. (a) If they play in the Super Bowl in Wisconsin and the weatherman predicts a \(60 \%\) chance of snow that day, what is the probability that San Francisco will win? (b) Given that San Francisco lost, what is the probability that the weather was bad?

Short answer

The probability that San Francisco will win is 40%. The probability that the weather was bad given that San Francisco lost is 80%.

Step-by-step solution

Define the Probabilities

Identify the given probabilities: - Probability of good weather: P(G) = 0.4 - Probability of bad weather: P(B) = 0.6 - Probability of winning in good weather: P(W|G) = 0.7 - Probability of winning in bad weather: P(W|B) = 0.2

Calculate Total Probability of Winning

Use the law of total probability to find the probability of winning regardless of the weather.\[ P(W) = P(W|G)P(G) + P(W|B)P(B) \] Plug in the values: \[ P(W) = 0.7 \times 0.4 + 0.2 \times 0.6 \] Simplify the equation: \[ P(W) = 0.28 + 0.12 = 0.40 \] Hence, the probability that San Francisco will win is 0.4 or 40%.

Determine Conditional Probability of Bad Weather Given a Loss

Use Bayes' theorem to find the probability of bad weather given that San Francisco lost: \[ P(B|L) = \frac{P(L|B)P(B)}{P(L)} \] Where: - P(L|B) = 1 - P(W|B) = 1 - 0.2 = 0.8- P(L) = 1 - P(W) = 1 - 0.4 = 0.6 Plug in the values: \[ P(B|L) = \frac{0.8 \times 0.6}{0.6} = 0.8 \] Hence, the probability that the weather was bad given that San Francisco lost is 0.8 or 80%.

Key concepts

Conditional Probability

Conditional probability helps us understand the likelihood of an event occurring given that another event has already happened. It is denoted as \( P(A|B) \) and is defined as the probability of event A occurring given that event B has occurred. In our football scenario, we see that the chance of San Francisco winning (W) changes based on the weather (B for bad and G for good). For instance, the probability of San Francisco winning given good weather is \(P(W|G) = 0.7\). This means there is a 70% chance they will win when the weather is good.
Understanding conditional probability is crucial, especially when dealing with interdependent events. It helps break down complex problems into manageable parts and gives a clearer picture of the situation.

Law of Total Probability

The law of total probability aids in finding the total probability of an outcome considering all possible scenarios. Mathematically, it is expressed as
\[ P(W) = P(W|G)P(G) + P(W|B)P(B) \] In the San Francisco game example, we consider both possible conditions: good and bad weather. The probability of winning overall is found by combining the probabilities of winning in each specific weather condition, weighted by the probability of each type of weather occurring. Hence,
\( P(W) = 0.7 \times 0.4 + 0.2 \times 0.6 = 0.40 \)
This tells us that San Francisco has a 40% total chance of winning, considering both weather conditions.

Bayes' Theorem

Bayes' theorem helps in finding the probability of a cause given an observed effect. It is extremely useful for updating the probability estimates as more evidence or information becomes available. The theorem is represented as:
\[ P(B|L) = \frac{P(L|B)P(B)}{P(L)} \]
Applying it to our football example, we determine the probability of bad weather (\

Statistical Thermodynamics

Statistical thermodynamics deals with the statistical behavior of systems with a large number of particles. It bridges the microscopic properties of particles with the macroscopic observable properties, such as temperature and pressure.
Although this concept appears distant from our football and weather scenario, the interconnectedness and dependability on various factors bear similarity. For example, predicting the overall performance of a football team can be likened to predicting the behavior of a gas in a container based on the motion of individual molecules.
It emphasizes how microscopic interactions can collectively shape the overall behavior, akin to how individual player performances and conditions like weather collectively affect the outcome of a game.