Problem 29 - Calculus for Biology and Medicine

Question

A box contains five red and four blue balls. You choose two balls. (a) How many possible selections contain exactly two red balls, how many exactly two blue balls, and how many exactly one of each color? (b) Show that the sum of the number of choices for the three cases in (a) is equal to the number of ways that you can select two balls out of the nine balls in the box.

Step-by-Step Solution

Answer

(a) 10 (two reds), 6 (two blues), 20 (one of each). (b) The sum, 36, matches the total combinations from nine balls.

Step 1: Calculate the Number of Ways to Choose Two Red Balls

The number of ways to choose two red balls from the five available is calculated using combinations. The formula for combinations is \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose. Thus, the number of ways to choose two red balls is \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \).

Step 2: Calculate the Number of Ways to Choose Two Blue Balls

We apply the same combinations formula to the four blue balls. The number of ways to choose two blue balls from four is \( \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \).

Step 3: Calculate the Number of Ways to Choose One Red Ball and One Blue Ball

To choose one red ball and one blue ball, we choose one ball from each color group. The number of ways to choose one red ball is \( \binom{5}{1} = 5 \) and to choose one blue ball is \( \binom{4}{1} = 4 \). Therefore, the number of ways to choose one red ball and one blue ball is \( 5 \times 4 = 20 \).

Step 4: Sum the Numbers from Steps 1, 2, and 3

Add the results from the previous steps to find the total number of possible selections. This gives us \( 10 + 6 + 20 = 36 \).

Step 5: Verify Total Number of Ways to Choose Any Two Balls

We calculate the total number of ways to choose any two balls from the total of nine balls using combinations. The total is \( \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \).

Step 6: Confirm That Calculations Match

We verify that the sum of the selections in the specific cases matches the total number of ways to choose any two balls. Both are equal to 36, confirming that the calculation is correct.

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. This concept is straightforward: it calculates the chance of something happening out of the total possibilities. In our ball example, you can view each draw and selection as a possible outcome. Calculating probability involves counting how many outcomes satisfy your condition and comparing it to the total number of possible outcomes. For example, if you want to know the probability of drawing two red balls, you first calculate how many ways you can pull out two red balls and then divide by the total ways you can draw any two balls. If there are 10 ways to draw two red balls and 36 ways to draw any two balls, the probability goes like this:

- Formula: Probability of event = (Number of favorable outcomes) / (Total number of outcomes) So in this specific case:
- Probability that both balls drawn are red = 10/36 Understanding probability helps predict outcomes based on mathematical reasoning, which makes you more aware of how likely or less likely something is in a given situation.

Combination formula

The combination formula is a mathematical tool used to determine the number of ways to choose a subset of items from a larger set, without regard to the order of selection. The formula for combinations, expressed as \( \binom{n}{r} \), is calculated by the expression \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \] where - \( n \) is the total number of items, - \( r \) is the number of items to choose, and - \( ! \) denotes a factorial, which is the product of an integer and all the integers below it (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)). In our example, when choosing two red balls from a total of five, the situation fits perfectly into the combination formula. You don't care about the order—that one red ball is picked first makes no difference compared to the other. This formula is handy for solving problems where arrangements don't matter, evoking efficiency and simplicity in tackling complexity. Knowing how to work with combinations changes how you approach and solve probability tasks.

Mathematical reasoning

Mathematical reasoning is the thought process of solving problems in a logical, sequential manner. It integrates understanding, rules, and the ability to conduct operations using numbers and abstract concepts. In the ball example, you use reasoning by methodically applying formulas to count all possibilities. Here's a stepwise approach to see reasoning in action:

• Identify the problem and what’s needed: We need to count how many ways to choose two balls of specific colors.
• Break it into smaller pieces: Calculate separately for each color combination—two reds, two blues, one of each.
• Use known formulas: Implement the combination formula for each scenario, \( \binom{5}{2} \) for red, \( \binom{4}{2} \) for blue, and \( 5 \times 4 \) for one of each.
• Compile: Add up all the different combinations to ensure no possibility is missed.
• Verify: Match your calculated total with an independent total selection calculation like \( \binom{9}{2} \).

This showcases both the verification process and consistency within mathematical reasoning, ensuring reliability in your solution. Reasoning builds confidence in handling not just mathematical problems but practical decisions beyond mathematics.