Problem 20 - Calculus for Biology and Medicine

Question

A drawer contains six bags numbered \(1-6\), respectively. Bag \(i\) contains \(i\) blue balls and 2 green balls. You roll a fair die and then pick a ball out of the bag with the number shown on the die. What is the probability that the ball is blue?

Step-by-Step Solution

Answer

The probability of drawing a blue ball is approximately 0.5278.

Step 1: Understand the Problem

We have a total of 6 bags, each corresponding to a die roll outcome from 1 to 6. Bag number \(i\) contains \(i\) blue balls and 2 green balls. We need to find the probability that a rolled die and the random selection from that bag results in a blue ball.

Step 2: Determine Bag Composition

Each bag \(i\) contains \(i\) blue balls and 2 green balls. Hence:- Bag 1 has 1 blue, 2 green (3 total)- Bag 2 has 2 blue, 2 green (4 total)- Bag 3 has 3 blue, 2 green (5 total)- Bag 4 has 4 blue, 2 green (6 total)- Bag 5 has 5 blue, 2 green (7 total)- Bag 6 has 6 blue, 2 green (8 total)

Step 3: Calculate Individual Bag Probabilities

To find the probability of drawing a blue ball from a particular bag \(i\), calculate\[P(\text{blue for bag } i) = \frac{i}{i + 2}\]Thus:- Bag 1: \( \frac{1}{3} \)- Bag 2: \( \frac{2}{4} = \frac{1}{2} \)- Bag 3: \( \frac{3}{5} \)- Bag 4: \( \frac{4}{6} = \frac{2}{3} \)- Bag 5: \( \frac{5}{7} \)- Bag 6: \( \frac{6}{8} = \frac{3}{4} \)

Step 4: Calculate the Overall Probability

The overall probability is the sum of the probabilities of each bag being chosen (\( \frac{1}{6} \) for each), multiplied by the probability of drawing a blue ball from that bag:\[P(\text{blue}) = \frac{1}{6}\left(\frac{1}{3}\right) + \frac{1}{6}\left(\frac{1}{2}\right) + \frac{1}{6}\left(\frac{3}{5}\right) + \frac{1}{6}\left(\frac{2}{3}\right) + \frac{1}{6}\left(\frac{5}{7}\right) + \frac{1}{6}\left(\frac{3}{4}\right)\]

Step 5: Simplify the Expression

Simplify each term:- \(\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}\)- \(\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\)- \(\frac{1}{6} \times \frac{3}{5} = \frac{1}{10}\)- \(\frac{1}{6} \times \frac{2}{3} = \frac{1}{9}\)- \(\frac{1}{6} \times \frac{5}{7} = \frac{5}{42}\)- \(\frac{1}{6} \times \frac{3}{4} = \frac{1}{8}\)

Step 6: Combine and Calculate

Add up all simplified probabilities:\[= \frac{1}{18} + \frac{1}{12} + \frac{1}{10} + \frac{1}{9} + \frac{5}{42} + \frac{1}{8}\]Convert each fraction to have a common denominator, or calculate directly with a calculator:The final probability is approximately 0.5278.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangement, and combination of objects. In this exercise, we utilized combinatorial principles to understand the different bag compositions after rolling a fair die. Each outcome of the die roll corresponds to one of the bags. Every bag, numbered from 1 to 6, is structured in a specific way based on the die result.

• Each bag contains a number of blue balls equal to its numerical designation and a fixed number of 2 green balls.
• Total number of balls in each bag is calculated as the sum of the blue and the green balls.

For example, bag 1 has 1 blue and 2 green balls, totaling 3 balls. Similarly, bag 2 contains 2 blue and 2 green balls, totaling 4. This combinatorial breakdown is crucial for determining probability outcomes that hinge on the selection from each bag. By listing out each potential composition, we set the foundation necessary for further probability calculations.

Conditional Probability

Conditional probability helps us in determining the likelihood of an event occurring, given that another event has occurred. In the context of our exercise, we needed to calculate the probability of selecting a blue ball given a particular bag number, which is chosen based on the dice roll outcome.

• The calculation for each individual bag involves dividing the number of blue balls by the total number of balls in the bag.
• This yields the probability of drawing a blue ball conditional on being in bag number i, denoted as \(P(\text{blue | bag } i) = \frac{i}{i+2}\).

This method allows us to focus specifically on the event happening under certain conditions, namely having already selected a specific bag. By calculating this for each bag, we gain insight into the behavior and nature of the problem under study.

Statistics

Statistics involves collecting, analyzing, presenting, and interpreting data. In probability exercises like this one, statistical methods aid in finding a comprehensive solution by combining individual probability events.

• Each bag of balls and their compositions give rise to associated probabilities.
• The use of statistical summation allows us to compute the overall probability of picking a blue ball in this scenario.

Here, we calculated the total probability by adding the individual probabilities of drawing a blue ball from each bag. Each probability was weighted by the chance of that particular bag being selected (\(\frac{1}{6}\)), since the die roll is fair. This combination of probabilities reflects the statistical principle of aggregation to achieve more complex probability outcomes from simpler individual events.