Question

PROBLEM SOLVING At a school, \(43 \%\) of students attend the homecoming football game. Only \(23 \%\) of students go to the game and the homecoming dance. What is the probability that a student who attends the football game also attends the dance?

Short answer

The probability that a student who attends the football game also attends the dance is approximately \(0.535\) or \(53.5\%\).

Step-by-step solution

Translate the problem into probabilities

Let's denote event \(F\) as the event a student attends the football game, and event \(D\) for the event a student attends the dance. From the problem, we know \(P(F) = 0.43\) (43% of students attend the football game) and \(P(F \cap D) = 0.23\) (23% of students attend both the football game and the dance).

Apply the formula for conditional probability

Mathematically, the conditional probability of an event \(D\), given that an event \(F\) has already occurred, is defined as \(P(D|F) = \frac{P(D \cap F)}{P(F)}\) . This formula allows us to determine the relative likelihood of event \(D\) given that \(F\) has occurred by comparing the proportion of times both \(D\) and \(F\) occur to the total number of times \(F\) occurs.

Compute the result with the given probabilities

Plugging the known values into the formula, we find \(P(D|F) = \frac{P(D \cap F)}{P(F)} = \frac{0.23}{0.43} = 0.53488372\).

Key concepts

Understanding Probability

Probability is a measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. For instance, if 43% of students attend the football game, this translates to a probability of 0.43 for any given student to attend the game.
Probability plays a crucial role in many areas of daily life and science, helping us to predict outcomes when faced with uncertainty.
To calculate probability, consider the ratio of the number of favorable outcomes to the total number of possible outcomes. In our example:

• The number of students attending the game is considered as favorable outcomes for event F.
• The total student population is the set of all possible outcomes.

Understanding this foundational concept is key to tackling more complex problems involving probability.

The Concept of Events Intersection

The intersection of events in the context of probability refers to the occurrence of two or more events at the same time. It is represented mathematically by the symbol \( \cap \). This is often referred to as joint probability.
In the given problem, we look at two events: attending the football game (event F) and attending the dance (event D). The notation \( P(F \cap D) = 0.23 \) represents the probability of students attending both events simultaneously.
Understanding intersections is vital for calculating conditional probabilities. It helps us assess how often two events happen together, which is a foundational aspect of probability theory.
When calculating joint probabilities, using the formula for conditional probability \( P(D|F) = \frac{P(D \cap F)}{P(F)} \), helps us understand the likelihood of event D given event F has occurred. This involves finding how many times both D and F happen out of the number of F occurrences.

Working with Fractions in Probability

Fractions are a central part of expressing and calculating probabilities. They represent proportions and probabilities require precise fractions to express the likelihood of an event. For instance, when the probability of an event is given as 43%, it equates to a fraction \( \frac{43}{100} \).
In the calculation of conditional probability, fractions allow us to express the part of a whole. Consider \( P(D|F) = \frac{P(D \cap F)}{P(F)} \), this formula involves division of probabilities which helps us see the proportional relationship.

• Numerator (\(P(D \cap F)\)): The intersection event as described earlier.
• Denominator (\(P(F)\)): The total occurrences of the event F.

Consistently working with fractions ensures that we calculate the true scale or proportion of the probability sought, adding precision to our problem-solving approach.