Question

MAKING AN ARGUMENT A meteorologist claims that there is a \(70 \%\) chance of rain. When it rains, there is a \(75 \%\) chance that your softball game will be rescheduled. Your friend believes the game is more likely to be rescheduled than played. Is your friend correct? Explain your reasoning.

Short answer

The friend's statement is correct. The probability of the game being rescheduled due to rain is \(52.5\%\), whereas the probability of the game being played (assuming the game only gets rescheduled if it rains) is \(47.5\%\), therefore it's more likely that the game will be rescheduled.

Step-by-step solution

Calculate the Probability of the Game Being Rescheduled

Given that the probability of it raining is \(70\%\), and if it rains, there is a \(75\%\) chance the game will be rescheduled. So, we multiply these two probabilities together to find the overall probability of the game being rescheduled due to rain. In other words, calculate \(0.70 \times 0.75 = 0.525\) (or \(52.5\%\) when expressed as a percentage).

Calculate the Probability of the Game Being Played

The probability of the game being played is the complementary event to the game being rescheduled. Therefore, we subtract the probability of the game being rescheduled from \(100\%\). Thus, calculate \(100\% - 52.5\% = 47.5\%\).

Compare the Probabilities and Conclude

Comparing the probabilities, we find that the probability of the game being rescheduled (\(52.5\%\)) is indeed higher than the probability of the game being played (\(47.5\%\)). Because the probability of the game being rescheduled is greater, the friend's statement is correct.

Key concepts

Complementary Events

In probability, complementary events are pairs of outcomes where one event happens if and only if the other does not. They are mutually exclusive, meaning they cannot both happen at the same time. For example, consider rolling a single six-sided die. The probability of rolling a 1 is a complementary event to the probability of not rolling a 1. When calculating probabilities, the sum of complementary events always equals 1, or 100% when expressed as a percentage. If you know the probability of one event, you can easily find its complement by subtracting the probability from 1.

• If the probability of an event \(A\) occurring is \(P(A)\), then its complement \(A^c\) not occurring is \(1 - P(A)\).
• In the context of the softball game, if we know there is a 52.5% chance of the game being rescheduled, the probability of the game not being rescheduled, or being played, is found using the complementary probability \(100\% - 52.5\% = 47.5\%\).

Understanding complementary events helps in many probability calculations, especially when you're trying to assess the likelihood of different outcomes.

Probability Calculations

Probability calculations involve determining how likely an event is to occur. This can be done by looking at the possible outcomes and determining how often an event happens out of the total number of possible outcomes. In scenarios where multiple independent events are involved, such as predicting the rescheduling of a game based on rain, probabilities are often multiplied.

• If event \(A\) has a probability \(P(A)\) of occurring, and event \(B\) has a probability \(P(B)\) of occurring after \(A\), the joint probability \(P(A \cap B)\) of both \(A\) and \(B\) occurring is computed as \(P(A) \times P(B)\).
• In the case of determining whether the softball game will be rescheduled, we multiplied the probability of rain (70% or 0.70) by the probability that the game is rescheduled given that it rains (75% or 0.75), resulting in a 52.5% probability.

Using these calculations helps in making informed decisions based on the likelihood of certain outcomes.

Meteorological Predictions

Meteorological predictions are forecasts about the weather and include estimating the probability of events like rain or storms. Meteorologists rely on data from satellites, radar, and weather models to make predictions.

• These forecasts express the likelihood of weather events as probabilities, such as a 70% chance of rain, which means there is a 70% likelihood that it will rain at a specific location during a certain time period.
• To make these predictions more actionable for the public, meteorologists often provide insights into the implications, like how weather conditions might affect scheduled activities, such as a softball game.

Understanding meteorological predictions allows people to better plan their activities and also helps in assessing risks based on weather forecasts. For instance, in our exercise, knowing the probability of rain helped assess the chance of the softball game being rescheduled.