Problem 13 - Algebra and Trigonometry

Question

\(13-14\) . A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) A red ball is drawn. (b) The ball drawn is not yellow. (c) A black ball is drawn.

Step-by-Step Solution

Answer

(a) \( \frac{5}{8} \); (b) \( \frac{7}{8} \); (c) 0.

Step 1: Understand the Total Number of Balls

First, calculate the total number of balls in the jar. There are 5 red balls, 2 white balls, and 1 yellow ball. Thus, the total number of balls is calculated as: \[ 5 + 2 + 1 = 8 \] So, there are 8 balls in total.

Step 2: Calculate Probability of Drawing a Red Ball

To find the probability of drawing a red ball, divide the number of red balls by the total number of balls:\[ P( ext{red}) = \frac{5}{8} \] Therefore, the probability of drawing a red ball is \( \frac{5}{8} \).

Step 3: Calculate Probability of Drawing a Non-Yellow Ball

To find the probability of drawing a ball that is not yellow, count the number of balls that are not yellow. There are 5 red and 2 white balls, totalling 7 non-yellow balls. Divide the number of non-yellow balls by the total:\[ P( ext{not yellow}) = \frac{7}{8} \] Thus, the probability of drawing a non-yellow ball is \( \frac{7}{8} \).

Step 4: Calculate Probability of Drawing a Black Ball

Since there are no black balls in the jar, the probability of drawing a black ball is:\[ P( ext{black}) = \frac{0}{8} = 0 \] Therefore, the probability of drawing a black ball is 0.

Key Concepts

Random Selection

Random selection is a foundational concept in probability. It refers to the process of making a choice without any specific pattern, logic, or bias.
In the context of drawing a ball from a jar, each ball has an equal chance of being selected when a random draw is performed.

This means every ball, regardless of its color, has a fair opportunity to be picked. The principle ensures that any specific ball has an unbiased probability based purely on its presence in the group.

• Imagine the jar as a mix of colors and each color representing a set quantity.
• When you choose without looking, you rely on random selection.

While it seems basic, mastering this concept is vital as it forms the basis for more complex probability calculations.

Event Probability

Event probability is about finding the likelihood of a specific outcome happening in an experiment.
It's calculated as the ratio of favorable outcomes to the total number of possible outcomes.

For instance, if you're figuring out the probability of drawing a red ball from a jar with 8 balls in total, you would count only the red balls out of the total.

• Here, there are 5 red balls out of 8, giving us a probability of \( \frac{5}{8} \).

This simple calculation can be applied to determine the event probability of other scenarios like drawing a non-yellow or a black ball.
The tricky part is identifying all possible outcomes accurately.

Color Probability

In problems like the one involving colored balls, color probability highlights how likely it is to pick a ball of a specific color.
Each color's probability depends on how many balls of that color there are compared to the whole set.

Understanding this concept is easier with examples:

• To find the probability of not picking a yellow ball, count the non-yellow balls (which are red and white in this case).
• Thus, we have 7 non-yellow balls out of a total of 8, leading to a probability of \( \frac{7}{8} \).

This shows how to calculate probability focusing on color, which also applies to any "what are the odds" type questions about objects differentiated by color.