Problem 30 - Algebra 2

Question

Determine whether the events are independent or dependent. Then find the probability. There are 3 miniature chocolate bars and 5 peanut butter cups in a candy dish. Judie chooses 2 of them at random. What is the probability that she chose 2 miniature chocolate bars?

Step-by-Step Solution

Answer

The events are dependent. Probability = \( \frac{3}{28} \).

Step 1: Identify the Events

Let's define the events. Event A is choosing the first miniature chocolate bar, and Event B is choosing the second miniature chocolate bar. We need to determine if these events are independent or dependent.

Step 2: Determine Independence or Dependence

Two events are independent if the outcome of one event does not affect the outcome of another. Here, once a miniature chocolate bar is chosen first, there are fewer chocolates in the dish for the second choice. Thus, Event B depends on Event A, making the events dependent.

Step 3: Calculate Total Outcomes

Calculate the total number of ways to choose any 2 candies from the 8 available candies. This is given by the combination formula \( C(n, k) \). Therefore, \( C(8, 2) = \frac{8 \times 7}{2 \times 1} = 28 \).

Step 4: Calculate Favorable Outcomes

Calculate the number of ways to choose 2 miniature chocolate bars from the available 3 bars. Using the combination formula again, \( C(3, 2) = \frac{3 \times 2}{2 \times 1} = 3 \).

Step 5: Find the Probability

The probability of Judie choosing 2 miniature chocolate bars is the ratio of favorable outcomes to total outcomes. Therefore, the probability is \( \frac{3}{28} \).

Key Concepts

Independent Events

In probability, independent events are those where the occurrence or outcome of one event does not influence the outcome of another event. This means that each event has its own separate probability, unaffected by the other event taking place.

For instance, imagine flipping a coin and rolling a die. The outcome of the coin flip (say, heads or tails) has no effect on the number you roll on the die (1 through 6). Each event stands alone in determining its result.

• The probability of independent events happening together is found by multiplying their individual probabilities.

Understanding independent events is crucial as it simplifies the calculation of their combined probabilities. However, remember that not all events are independent.
It’s essential to analyze each scenario to determine whether events influence each other or not.

Dependent Events

Dependent events occur when the outcome of one event affects the outcome of a second event. This is common in situations where the events are connected or when items are removed without replacement, as was the case in the exercise with Judie's candies.

Once Judie selects one chocolate bar, the number of available candies for the next choice changes, making the events dependent. This dependency requires us to adjust the probability calculations to reflect the reduced number of choices.

• For dependent events, the probability of both occurring can be found by multiplying the probability of the first event with the probability of the second event happening after the first.

Recognizing dependencies between events helps in accurately determining probabilities, as seen in Judie’s example. It's important to identify these dependencies to avoid mistakes in calculations.

Combination Formula

The combination formula is a powerful tool used in probability to determine how many ways we can select a certain number of items from a larger pool without considering the order of selection. The formula is written as \( C(n, k) \), where \( n \) is the total number of items, and \( k \) is the number of items to choose.

The mathematical expression for the combination formula is:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \], where \( ! \) denotes factorial, which is the product of all positive integers up to that number.

In Judie’s problem, we used this formula twice:

• To find the total number of ways to select any 2 candies from 8 candies (\( C(8, 2) = 28 \)).
• To find the number of ways to select 2 chocolate bars from the 3 available (\( C(3, 2) = 3 \)).

Understanding how to use the combination formula is essential for solving problems that involve choosing items from a set, especially in probability, where order does not matter.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred, which is crucial when dealing with dependent events.

In Judie's example, the probability of choosing the second chocolate bar changes based on the fact that the first chocolate bar has already been chosen. Knowing one condition allows us to adjust the probability of the next event accordingly.

The formula for conditional probability is given as:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

• Where \( P(B|A) \) is the probability of event B occurring given that A has occurred.
• \( P(A \cap B) \) is the likelihood of both events A and B occurring.
• \( P(A) \) is the probability of event A occurring alone.

Applying conditional probability ensures a more accurate calculation by taking into account the effect of prior events on upcoming outcomes, which is essential when events are dependent.