Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. a. Describe an appropriate sample space for this experiment. b. Describe the event "a head is tossed and an even number is rolled."

Step-by-Step Solution

Answer

a. The appropriate sample space for this experiment is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. b. The event "a head is tossed and an even number is rolled" can be described as the subset of the sample space containing the outcomes {H2, H4, H6}.

Step 1: Identify possible outcomes for each action separately.

For the coin, there are two possible outcomes: heads (H) and tails (T). For the die, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

Step 2: Determine all possible outcomes of the experiment.

To find the appropriate sample space for this experiment, we need to consider all possible outcomes resulting from tossing a coin and rolling a die. This can be done with the help of a table: ``` Coin\Dices | 1 | 2 | 3 | 4 | 5 | 6 H | H1| H2| H3| H4| H5| H6 T | T1| T2| T3| T4| T5| T6 ``` The sample space S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Step 3: Identify the outcomes in the event "a head is tossed and an even number is rolled."

To find the outcomes for the event "a head is tossed and an even number is rolled," we need to look for outcomes that have a head on the coin AND an even number on the die. In the sample space, the outcomes that meet these conditions are: H2, H4 and H6.

Step 4: Describe the event using the identified outcomes.

The event "a head is tossed and an even number is rolled" can be described as the subset of the sample space containing the outcomes {H2, H4, H6}.

Key Concepts

Sample Space

In probability, the sample space is the set of all possible outcomes of an experiment. Think of it like the complete gallery of every potential scenario that could unfold. For example, when we toss a coin and roll a die, we want to capture every result this experiment can produce.

• Tossing a coin can yield two outcomes: heads (H) or tails (T).
• Rolling a die can result in any of six outcomes: 1, 2, 3, 4, 5, or 6.

To create the sample space for this combined experiment, we pair each result from the coin with every result from the die.
The sample space then will contain combinations like H1, H2, T1, T2, and so on, totaling twelve unique outcomes. A visual method to help is to imagine a grid or table where each row and column intersects at one of these result combinations.

Independent Events

Two events are independent if the outcome of one doesn't affect the probability of the other. Picture each event occurring in its own bubble, unaffected by what happens with the other bubble.
With our example, the action of tossing the coin is independent from rolling the die. Whether the coin shows heads or tails, the number that shows up on the die remains unaffected.

• Flipping the coin to show H or T.
• Rolling any number on the die from 1 to 6.

Each outcome is as likely to occur as if the other action hadn't taken place. This concept of independence greatly simplifies calculating probabilities because we can examine the events separately without concern of crossover influences between them.

Outcomes

Outcomes in probability represent the results of an experiment. In our scenario, each outcome is a specific combination of the coin toss and the die roll.
For example, if you toss the coin and get a head, and roll the die to get a 4, the outcome is recorded as H4. Each outcome reveals something distinct about the experiment you performed.

• Outcome H1: Head from the coin, 1 from the die.
• Outcome T3: Tail from the coin, 3 from the die.

Within the sample space, every possible outcome must be accounted for. Recognizing different outcomes helps us understand what scenarios are included in certain events, like observing a head and an even number, which in our example translates to H2, H4, and H6.

Experiments in Probability

An experiment in probability explores potential outcomes through a specific procedure. It's the action you perform to gain insight into various occurrences and the likelihood of their outcome.
In our case, the experiment involves two parts: tossing a coin and rolling a die. Each time we perform this set of actions, we observe the outcomes to better understand probabilities.

• Before experimenting: Define the sample space so you know all potential outcomes.
• During experimenting: Identify new outcomes as they happen and see how they fit within your defined sample space.

Grasping the details of a probability experiment means you'll be equipped to tackle questions about likelihood, such as how often a head and an even number appear based on our criteria above.