Problem 8 - Algebra and Trigonometry

Question

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) Neither a white nor yellow ball is drawn. (b) A red, white, or yellow ball is drawn. (c) The ball drawn is not white.

Step-by-Step Solution

Answer

(a) \(\frac{5}{8}\); (b) 1; (c) \(\frac{3}{4}\).

Step 1: Identify Total Outcomes

The jar contains a total of 5 red balls, 2 white balls, and 1 yellow ball. To find the probability of each event, we first need to determine the total number of balls in the jar. We add the numbers of each color together: 5 red + 2 white + 1 yellow = 8 balls. Therefore, there are 8 total outcomes.

Step 2: Event (a): Calculate Probability of Drawing Neither White Nor Yellow

The probability of not drawing a white or yellow ball is the probability of drawing a red ball (since red is the only other option). There are 5 red balls, so the probability is the number of favorable outcomes (5 red balls) over the total number of outcomes (8 balls), \[P(\text{not white or yellow}) = \frac{5}{8}.\]

Step 3: Event (b): Calculate Probability of Drawing Red, White, or Yellow

The event of drawing a red, white, or yellow ball encompasses all balls in the jar since these are the only colors available. Thus, the probability of drawing any ball (red, white, or yellow) is 1, \[P(\text{red, white, or yellow}) = \frac{8}{8} = 1.\]

Step 4: Event (c): Calculate Probability of Drawing a Non-White Ball

A non-white ball can either be red or yellow. There are 5 red balls and 1 yellow ball, making a total of 6 balls that are not white. Therefore, the probability of drawing a non-white ball is \[P(\text{not white}) = \frac{6}{8} = \frac{3}{4}.\] Simplifying this gives us 0.75.

Key Concepts

Probability of Events

Probability is a fundamental concept in statistics and mathematics that measures the likelihood of a particular event occurring. It ranges from 0 to 1, where 0 indicates impossibility and 1 indicates certainty. In the context of drawing balls from a jar, we are interested in calculating the probability of specific events.

• To calculate probability, we use the formula: \[P( ext{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}\]
• The total number of possible outcomes is the sum of all the different types of balls present in the jar, which is 8 in this case.
• By identifying the favorable outcomes for each scenario, we can determine the probability of each event.

For example, the probability of neither drawing a white nor a yellow ball is calculated using the number of red balls, as those are the only other options. Understanding these probabilities helps us predict outcomes in uncertain situations.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting combinations and permutations of objects. It is particularly useful in probability, as it aids in understanding how combinations of outcomes can develop.

• When calculating probabilities, the arrangement or selection of balls matters because each represents a different event or outcome.
• The exercise asks us to consider combinations, such as drawing neither a white nor a yellow ball, which involves excluding certain outcomes to find the probability.
• Understanding the total and favorable outcomes is a combinatorial technique that simplifies probability calculations.

These concepts are essential when analyzing complex scenarios with multiple possibilities, such as finding the likelihood of specific sequences of events.

Basic Algebra Concepts

Basic algebra concepts lay the groundwork for solving probability problems. Algebra involves using formulas and equations to find unknown values, and these skills are pivotal when computing probabilities.

• In our example of drawing balls, basic algebra helps formalize the process of calculating probabilities by defining the relationships between favorable and total outcomes.
• Simplifying fractions, such as converting \( \frac{6}{8} \) to \( \frac{3}{4} \), is an application of algebraic skills.
• Algebra also helps in checking if probabilities are reasonable by ensuring that the total probability of all possible events adds up to 1.

By applying basic algebra to probability scenarios, students can accurately determine the likelihood of events and gain deeper insights into the relationships between different outcomes.