Problem 23 - Algebra and Trigonometry

Question

Two balls are picked at random from a jar that contains three red and five white balls. Find the probability of the following events. (a) Both balls are red. (b) Both balls are white.

Step-by-Step Solution

Answer

(a) Probability of two red balls: 0.107 (b) Probability of two white balls: 0.357

Step 1: Understand the Problem

The problem involves picking two balls from a jar that contains three red balls and five white balls. We need to calculate the probability of picking two red balls (part a) and the probability of picking two white balls (part b).

Step 2: Calculate Total Number of Ways to Pick Two Balls

First, calculate the total number of ways to pick any two balls from the eight balls in the jar. We use the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. So, for picking two balls from eight, it is \( \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \).

Step 3: Calculate Ways to Pick Two Red Balls

Calculate the number of ways to pick two red balls from the three available red balls. We use the combination formula \( \binom{3}{2} \). Thus, \( \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3 \).

Step 4: Calculate the Probability Both Balls are Red

The probability of picking two red balls is the ratio of the number of ways to pick two red balls to the total number of ways to pick any two balls. Thus, the probability is \( \frac{3}{28} \approx 0.107 \).

Step 5: Calculate Ways to Pick Two White Balls

Calculate the number of ways to pick two white balls from the five available white balls. We use the combination formula \( \binom{5}{2} \). Thus, \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \).

Step 6: Calculate the Probability Both Balls are White

The probability of picking two white balls is the ratio of the number of ways to pick two white balls to the total number of ways to pick any two balls. Thus, the probability is \( \frac{10}{28} = \frac{5}{14} \approx 0.357 \).

Key Concepts

Combination Formula

In the world of probability and counting, understanding how to calculate combinations is essential. A combination is a selection of items from a larger set, where the order of selection does not matter. This is different from permutations, where order does matter.

We use the combination formula, denoted as \( \binom{n}{k} \), where \( n \) represents the total number of items, and \( k \) is the number of items to select. The formula itself is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

The exclamation mark denotes a factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In the given exercise, to find how many ways two balls can be chosen from eight, we apply this formula: \( \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \). Similarly, combinations are used to calculate the number of ways to pick two red balls from three and two white balls from five, using \( \binom{3}{2} = 3 \) and \( \binom{5}{2} = 10 \) respectively.

Probability Calculation

Probability is a measure of the likelihood that a given event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In mathematical terms:

• Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

Probability values range from 0 (impossible event) to 1 (certain event).

In the exercise, the probability of both balls being red is calculated by finding how many ways 2 red balls can be selected out of 3, compared to how two balls can be chosen out of 8 overall. Therefore, the probability is \( \frac{3}{28} \approx 0.107 \).Similarly, for two white balls, you calculate the probability by comparing the ways to pick two white balls out of five with choosing any two from eight total. This gives us a probability of \( \frac{10}{28} = \frac{5}{14} \approx 0.357 \).

These calculations demonstrate how understanding both the total possibilities and the desired outcomes helps us determine the probability of an event.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with discrete elements that uses algebra and arithmetic and is primarily concerned with countable, distinct items. Unlike continuous mathematics, it does not involve concepts of things that are not countable or are ambiguous.

Key areas of focus in discrete mathematics include combinatorics, graph theory, and logic. The combination formula and probability calculation methods used in the exercise are foundational aspects of combinatorics—a major area within discrete mathematics. These methods help us solve problems involving finite structures such as, in this case, counting the ways to combine red and white balls from a finite set.

• Combinatorics allows us to count possible configurations.
• It provides tools to calculate probabilities in scenarios where outcomes are discrete, distinct, and countable.

Understanding how discrete mathematics underpins these calculations helps students tackle a range of problems, from simple probability challenges to more complex real-world applications like coding theory and network design.