Problem 4 - Calculus for Biology and Medicine

Question

Determine the sample space for each random experiment. An urn contains six balls numbered \(1-6\), respectively. The random experiment consists of selecting five balls simultaneously without replacement.

Step-by-Step Solution

Answer

The sample space is \(\{\{1,2,3,4,5\}, \{1,2,3,4,6\}, \{1,2,3,5,6\}, \{1,2,4,5,6\}, \{1,3,4,5,6\}, \{2,3,4,5,6\}\}\).

Step 1: Understand the Problem

The problem involves finding the sample space for a random experiment. The experiment described is selecting five balls from an urn containing six balls numbered 1 through 6 without replacement.

Step 2: Define the Sample Space

The sample space of a random experiment is the set of all possible outcomes. For this experiment, each possible outcome is an unordered combination of 5 balls chosen from the 6 available balls in the urn.

Step 3: Calculate the Number of Combinations

To calculate the number of combinations of choosing 5 balls from 6, we use the combination formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]where \(n\) is the total number of balls, which is 6, and \(k\) is the number of balls selected, which is 5.

Step 4: Apply the Combination Formula

Substitute the values into the formula:\[\binom{6}{5} = \frac{6!}{5!(6-5)!} = \frac{6!}{5! \times 1!} = 6\]This calculation indicates that there are 6 unique combinations of 5 balls that can be chosen from the 6 balls.

Step 5: List the Sample Space

The sample space consists of all the unique combinations calculated: \(\{ \{1,2,3,4,5\}, \{1,2,3,4,6\}, \{1,2,3,5,6\}, \{1,2,4,5,6\}, \{1,3,4,5,6\}, \{2,3,4,5,6\} \}\). These are the 6 distinct combinations possible from choosing 5 out of 6 balls.

Key Concepts

Random Experiment

In probability theory, a random experiment is an essential concept that refers to a process or action that leads to one of several possible outcomes. These outcomes cannot be predicted with certainty, but the way the experiment is conducted is known beforehand. For instance, in our given exercise of selecting balls from an urn, the random experiment is the act of choosing 5 balls simultaneously from a set of 6.

This process is considered 'random' because we do not know precisely which combination of balls will be picked each time the experiment is conducted. It is the randomness and the unpredictable nature of the outcome that qualifies an event as a random experiment.

Some characteristics of random experiments include:

• The set of all possible outcomes is predefined.
• The outcome of a single trial cannot be determined beforehand.
• The process can be repeated under the same conditions.

Understanding random experiments is crucial for computing probabilities and defining sample spaces in probability theory.

Combinations

Combinations are a vital mathematical concept used to determine how many different ways a set of items can be selected from a larger set. In the context of probability and our exercise, a combination refers to the selection of 5 balls out of 6 without regard to order.

The idea here is that the order in which the balls are selected does not matter, meaning picking ball 1, then ball 2, is the same as picking ball 2, then ball 1.

The formula to calculate combinations is given by:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where:

• \(n\) is the total number of items to choose from.
• \(k\) is the number of items to select.
• \(!\) denotes factorial, the product of all positive integers up to that number.

In the exercise, we use this formula to determine that there are 6 different combinations of choosing 5 balls from 6, highlighting that combinations are fundamental when it comes to defining sample spaces.

Probability

Probability is the measure of the likelihood that a particular outcome will occur in a random experiment. It is commonly expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

In the scenario of our exercise, understanding the probability helps us assess how likely it is to draw each unique set of 5 balls from the urn.

Here are some key points about probability:

• The probability of all possible outcomes in a sample space adds up to 1.
• The probability of an individual event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
• When each outcome in a sample space is equally likely, the probability of an event is simplified to the ratio of successful outcomes to total outcomes.

In our exercise, since each of the 6 combinations is equally possible, each has a probability of \(\frac{1}{6}\) if picking any one combination is our event of interest. Understanding probability allows you to predict and analyze the possible outcomes of random experiments.