Problem 22 - Algebra 2

Question

For Exercises \(20-23,\) determine whether the events are mutually exclusive or inclusive. Then find the probability. In the Math Club, 7 of the 20 girls are seniors, and 4 of the 14 boys are seniors. What is the probability of randomly selecting a boy or a senior to represent the Math Club at a statewide math contest?

Step-by-Step Solution

Answer

The events are inclusive, and the probability is \( \frac{21}{34} \).

Step 1: Understand the Events

We have two events to consider. Event A is selecting a boy, and Event B is selecting a senior. A boy or a senior can be selected, so we need to determine if these events overlap (inclusive) or do not overlap (mutually exclusive).

Step 2: Define the Events Mathematically

Let Event A be selecting a boy and Event B be selecting a senior. We have 14 boys out of 34 students and 11 seniors (7 girls + 4 boys) out of 34 students. The total number of students is 34, which is the sum of boys and girls in the club.

Step 3: Determine if Events are Mutually Exclusive or Inclusive

An event is mutually exclusive if both events cannot happen at the same time. Since some boys are also seniors, the events are inclusive.

Step 4: Calculate Probability of Event A

The probability of selecting a boy is the ratio of boys to total students. So, \( P(A) = \frac{14}{34} \).

Step 5: Calculate Probability of Event B

The probability of selecting a senior is the ratio of seniors to total students. So, \( P(B) = \frac{11}{34} \).

Step 6: Calculate Probability of Intersection of Events

Calculate the probability of selecting a boy who is also a senior, which is \( P(A \cap B) = \frac{4}{34} \) because there are 4 senior boys.

Step 7: Apply the Inclusion-Exclusion Principle

For inclusive events, the probability of selecting a boy or a senior is given by the formula:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]Plugging in our values:\[ P(A \cup B) = \frac{14}{34} + \frac{11}{34} - \frac{4}{34} = \frac{21}{34} \]

Step 8: Simplify the Probability

The probability \( \frac{21}{34} \) is already in its simplest form. It represents the probability of selecting a boy or a senior.

Key Concepts

Mutually Exclusive Events

In probability, events are termed mutually exclusive if they cannot occur at the same time. This means that the occurrence of one event inherently prevents the occurrence of the other. A classic example is flipping a coin, where getting heads knocks out the possibility of getting tails in the same flip. For mutually exclusive events, the probability of either event occurring is the sum of their individual probabilities. Mathematically, for two events, A and B, this is represented by:

• \( P(A \cup B) = P(A) + P(B) \)

However, in our exercise example, since some boys are also seniors, the events of selecting a boy and selecting a senior are not mutually exclusive, as they can happen simultaneously.

Inclusive Events

When two events can occur simultaneously, they are known as inclusive events. This overlap means that both events can happen at once, which is the case when selecting a boy or a senior from the Math Club, as some boys are also seniors. In such cases, calculating the probability directly by simply adding each event’s probability would lead to double-counting the overlapping part.
To adjust for this overlap, we leverage the inclusion-exclusion principle, ensuring our calculations account for this shared occurrence accurately. Thus, recognizing inclusive events is crucial in probability calculation to avoid errors in overestimating the probabilities involved.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is a helpful method in probability that adjusts for double-counting when dealing with inclusive events. Essentially, you calculate the probability of at least one of the events occurring by summing their individual probabilities and then subtracting the probability of both events happening at the same time.

• The formula is:\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

In our exercise, this helps us correctly find the probability of selecting either a boy or a senior without counting senior boys twice. This principle ensures our calculations are precise and reflect the real likelihood of either event occurring.

Probability Calculation

Calculating probability involves determining how likely it is for a specific event to happen among all possible outcomes. The probability value ranges from 0 to 1, where 0 implies impossibility and 1 indicates certainty.
To compute the probability of an event, divide the number of successful outcomes by the total number of possible outcomes. For example, in the Math Club exercise, you find the probability of randomly selecting a boy by dividing the number of boys (14) by the total number of students (34):

• \( P(A) = \frac{14}{34} \)

The formula for inclusive events using the inclusion-exclusion principle ensures you assess the correct likelihood when events overlap, as seen in our example where both boys and seniors are counted. Accurate probability calculations are foundational in statistical reasoning and decision-making processes.