Problem 29 - Algebra 2

Question

REVIEW Your gym teacher is randomly distributing 15 red dodge balls and 10 yellow dodge balls. What is the probability that the first ball that she hands out will be yellow and the second will be red? $$ \begin{array}{ll}{\mathrm{F} \frac{1}{24}} & {\mathrm{H} \frac{2}{5}} \\\ {\mathrm{G} \frac{1}{4}} & {\mathrm{J} \frac{23}{25}}\end{array} $$

Step-by-Step Solution

Answer

The correct probability is \( \frac{1}{4} \), option G.

Step 1: Understand the Total Number of Balls

The gym teacher has a total of 15 red dodge balls and 10 yellow dodge balls, making a total of 25 balls. This is important to understand as it helps to calculate the probability.

Step 2: Calculate Probability of First Yellow Ball

The probability of picking a yellow ball first is calculated by dividing the number of yellow balls by the total number of balls: \( P( ext{First Yellow}) = \frac{10}{25} = \frac{2}{5} \).

Step 3: Calculate Probability of Second Red Ball Given First Yellow

After one yellow ball is picked, there are 24 balls left with 15 of them being red. The probability of picking a red ball next is: \( P( ext{Second Red | First Yellow}) = \frac{15}{24} = \frac{5}{8} \).

Step 4: Calculate Combined Probability

The overall probability of picking a yellow ball first and then a red ball is the product of the individual probabilities: \[ P( ext{First Yellow and Second Red}) = \frac{2}{5} \times \frac{5}{8} = \frac{2}{8} = \frac{1}{4} \].

Step 5: Compare with Provided Options

The calculated probability is \( \frac{1}{4} \), which corresponds to option G. Therefore, \( \frac{1}{4} \) is the correct answer.

Key Concepts

Combined Probability

Combined probability is the probability of two or more events occurring together. In the exercise with dodge balls, we want to find the likelihood that two specific events will happen: a yellow ball is chosen first, followed by a red ball. We calculate this combined probability by multiplying the probability of each independent event.

To find the combined probability, you need to:

• Determine the probability of the first event happening, which in this exercise is choosing a yellow ball. With 10 yellow balls out of 25, this probability is \( \frac{2}{5} \).
• Next, find the probability of the second event happening after the first one. This means, picking a red ball after a yellow one has already been picked. The probability for this is \( \frac{5}{8} \).
• Multiply these probabilities to find the combined probability: \( \frac{2}{5} \times \frac{5}{8} = \frac{1}{4} \).

The product gives us the overall likelihood of both events occurring in sequence.

Conditional Probability

Conditional probability involves finding the probability of an event given that another related event has already happened. In this exercise, after understanding the total number of balls and picking a yellow one first, we need to calculate the probability of picking a red ball second.
Here’s a step-by-step look at conditional probability:

• Start by acknowledging that the first event (picking a yellow ball) changes the conditions. Instead of having 25 balls, you now have only 24 left.
• With the red balls remaining unchanged at 15, the probability formula for this conditional event becomes \( P(\text{Second Red | First Yellow}) = \frac{15}{24} \).
• Simplifying \( \frac{15}{24} \) gives \( \frac{5}{8} \).

This calculated probability is crucial as it provides the likelihood of the second event given the first event has occurred.

Random Events

In probability, a random event is any occurrence that can happen with an uncertain outcome. Each event in our gym teacher's distribution scenario is a random event. The selection process for each dodge ball is completely random.
Some characteristics of random events to consider:

• Random events are described by their probability, which quantifies the likelihood of their occurrence.
• In this scenario, each time the teacher picks a ball, it is an independent event with its probability recalculated, especially after the first choice alters the total number.
• The randomness ensures there is no bias in the selection process.

Understanding these characteristics is vital as it underscores why the order and the portion of remaining balls matter in calculating probabilities.