Question

Two cold tablets are unintentionally placed in a box containing two aspirin tablets. The four tablets are identical in appearance. One tablet is selected at random from the box and is swallowed by the first patient. A tablet is then selected at random from the three remaining tablets and is swallowed by the second patient. Define the following events as specific collections of simple events: a. The sample space \(S\) b. The event \(A\) that the first patient obtained a cold tablet c. The event \(B\) that exactly one of the two patients obtained a cold tablet d. The event \(C\) that neither patient obtained a cold tablet

Short answer

Answer: The events representing the cases where at least one patient obtained a cold tablet are events A and B. In event A, the first patient obtained a cold tablet, and in event B, exactly one of the two patients obtained a cold tablet.

Step-by-step solution

Identify the outcomes of the sample space

Since we have two cold tablets and two aspirin tablets, let's represent them as follows: C1, C2, A1, and A2. We will find all the possible combinations of tablets taken by the first patient and second patient.

List all possible combinations for the sample space

The possible combinations of tablets taken by first (F) and second (S) patients are (F - S): (C1-A1), (C1-A2), (C1-C2), (A1-A2), (A1-C1), (A1-C2), (A2-C1), (A2-C2). So the sample space \(S\) is: $$ S = \{(C1-A1), (C1-A2), (C1-C2), (A1-A2), (A1-C1), (A1-C2), (A2-C1), (A2-C2)\} $$

Define event A, the first patient obtained a cold tablet

Identify the outcomes in which the first patient had a cold tablet. These outcomes are: (C1-A1), (C1-A2), (C1-C2). So event \(A\) is: $$ A = \{(C1-A1), (C1-A2), (C1-C2)\} $$

Define event B, exactly one of the two patients obtained a cold tablet

Identify the outcomes in which exactly one of the patients had a cold tablet. These outcomes are: (C1-A1), (C1-A2), (A1-C1), (A2-C1). So event \(B\) is: $$ B = \{(C1-A1), (C1-A2), (A1-C1), (A2-C1)\} $$

Define event C, neither patient obtained a cold tablet

Identify the outcomes in which neither patient had a cold tablet. The only outcome with no cold tablets is (A1-A2). So event \(C\) is: $$ C = \{(A1-A2)\} $$ In conclusion, we have defined the sample space \(S\), events \(A\), \(B\), and \(C\) according to the given scenario. This will help us analyze possible scenarios and calculate probabilities based on the defined events.

Key concepts

Sample Space

In probability, the sample space refers to the set of all possible outcomes of a particular experiment. It is the foundation on which probability theory is built. Whenever you perform an experiment or observe a process, there is a list of all the possible results or configurations that could occur. This list collectively is called the sample space.

From our exercise, the sample space is comprised of all combinations of two tablets, consisting of a mix of cold and aspirin tablets, selected in turn by two patients. We use the labels C1, C2 for cold tablets and A1, A2 for aspirin tablets. The sample space is therefore:

• (C1-A1)
• (C1-A2)
• (C1-C2)
• (A1-A2)
• (A1-C1)
• (A1-C2)
• (A2-C1)
• (A2-C2)

These combinations represent every possible pairing of tablets that a first and then a second patient might take. Essentially, the sample space allows us to see all possible outcomes of the tablet selection process.

Random Selection

Random selection is a fundamental concept in probability which ensures that each possible outcome is equally likely. It's like drawing names from a hat without looking; each name has an equal chance of being drawn.

In the exercise's scenario, random selection applies when each patient picks a tablet. There is no bias or predetermined order in which the tablets are chosen. The tablets are identical in appearance, which means that both patients have an equal opportunity to pick either a cold or an aspirin tablet. This randomness is crucial because it allows us to apply probability theory and calculate the likelihood of different events occurring.

Events in Probability

In probability, an event is a specific collection of outcomes from the sample space. Think of events as the interesting occurrences or scenarios that we might want to focus on within all the possible outcomes listed in the sample space. Events can contain one, some, or all the outcomes in the sample space.

In our exercise example, we define specific events related to tablet selection:

• Event A: The first patient receives a cold tablet. This event includes outcomes like (C1-A1), (C1-A2), and (C1-C2).
• Event B: Exactly one patient receives a cold tablet. Outcomes meeting this criteria are (C1-A1), (C1-A2), (A1-C1), and (A2-C1).
• Event C: Neither patient receives a cold tablet. The only outcome for this is (A1-A2).

Understanding events helps us determine which specific outcomes or combination of outcomes we are interested in analyzing further or calculating probabilities for.

Probability of Events

The probability of an event tells us how likely that event is to occur, expressed as a fraction, decimal, or percentage. It is calculated by dividing the number of favorable outcomes for the event by the total number of possible outcomes in the sample space.

In our exercise, if we want to find the probability that one of the events occurs, we count how many ways that event can happen and then divide by the total number of outcomes. For instance:

• The probability of event A (first patient gets a cold tablet) is the number of outcomes in A divided by the total number of outcomes in the sample space, which is 3/8 since there are 3 favorable outcomes: (C1-A1), (C1-A2), (C1-C2).
• The probability of event B (exactly one patient gets a cold tablet) would be 4/8 or 1/2, as there are 4 favorable outcomes out of 8.
• Finally, the probability of event C (neither patient gets a cold tablet) is 1/8 since only (A1-A2) fits this criteria.

By understanding the probability of different events, one can make informed predictions or decisions based on the likelihood of various outcomes.