Question

Define the simple events for the experiments in Exercises \(16-20 .\) Three children are randomly selected and their gender is recorded.

Short answer

Answer: The simple events in this experiment are: - Simple Event 1 (E1): {MMM} - Simple Event 2 (E2): {MMF} - Simple Event 3 (E3): {MFM} - Simple Event 4 (E4): {MFF} - Simple Event 5 (E5): {FMF} - Simple Event 6 (E6): {FMM} - Simple Event 7 (E7): {FMМ} - Simple Event 8 (E8): {FFF}

Step-by-step solution

Determine possible gender combinations

In this experiment, there are two possible outcomes for each child's gender: Male or Female. Let's denote Male as "M" and Female as "F". Since we are selecting three children, we can create possible combinations of their genders.

List all possible combinations

With two possible outcomes for each child and three children, we can calculate the total number of combinations using 2^(3) = 8 combinations. Now, we'll list out all those combinations: 1. MMM 2. MMF 3. MFM 4. MFF 5. FMF 6. FMМ 7. FMM 8. FFF

Define the simple events

After determining all possible gender combinations for the three children, we can define the simple events for this experiment. A simple event in this context is the possible outcome of selecting three children and recording their genders: - Simple Event 1 (E1): {MMM} - Simple Event 2 (E2): {MMF} - Simple Event 3 (E3): {MFM} - Simple Event 4 (E4): {MFF} - Simple Event 5 (E5): {FMF} - Simple Event 6 (E6): {FMM} - Simple Event 7 (E7): {FMМ} - Simple Event 8 (E8): {FFF} These simple events can be used to further analyze the probabilities and outcomes of this experiment.

Key concepts

Probability Combinations

Understanding probability combinations is fundamental when we're predicting or analyzing real-life situations. Let's consider a situation where we are trying to determine all the possible gender combinations of three randomly selected children. This scenario treats each selection as independent, meaning the gender of one child does not influence the gender of the others. Since there are two potential outcomes for gender (Male or Female), and each child can be either, we inherently work with a concept known as permutations and combinations, a pillar of probability theory.

To enumerate these, one handy method is to use the powers of the number of choices per child. Specifically, with two possible outcomes (M or F) and three children, the total number of combinations is given by the mathematical expression: \(2^3 = 8\). This formula implies that there are eight possible ways these genders can be combined. You can visualize or list them out systematically, ensuring you cover all the possibilities like so:

• MMM (all males)
• MMF
• MFM
• MFF
• FMF
• FMM
• FFM
• FFF (all females)

Each of these gender combinations represents a unique scenario, showcasing the power of probability combinations in quantifying potential outcomes in simple events.

Gender Probability

Gender probability typically deals with the likelihood of the occurrence of a particular gender in a given scenario. In our example of selecting three children, we want to determine the probability of getting different gender combinations. Let's assume that the probability of selecting a male or female child is equally likely, seeing as there’s no information provided to suggest otherwise.

Counting Individual Scenarios

For instance, to calculate the probability of getting the combination of two males and one female ('MMF', 'MFM', or 'FMM'), we would consider each of these a favorable outcome. Since there are three such combinations out of eight total possibilities, we would say the probability is \(\frac{3}{8}\) for that specific event.

However, we must also remember that real-life gender ratios may not always be equally distributed, and probabilities can shift when additional information or assumptions alter the likelihood of selecting either gender. When studying probability, it’s crucial to recognize these nuances and understand how they influence outcomes.

Probability Outcomes

Probability outcomes refer to all the possible results of a random process. In the context of our exercise, they tie back to the notion of simple events. So when we're looking at a family with three children, we've already established the eight possible outcomes in terms of their gender. However, delving deeper, each of these outcomes has an associated probability.

If we assume each gender has an equal chance of being chosen, each child represents a coin flip, with 'heads' perhaps being male and 'tails' female. Under this assumption, each of the eight simple events - 'MMM', 'MMF', 'MFM', 'MFF', 'FMF', 'FMM', 'FFM', and 'FFF' — would have an identical probability of \(\frac{1}{8}\) because all outcomes are equally likely in this scenario.

Visualizing with a Tree Diagram

To further aid understanding, a tree diagram is a fantastic visualization tool used in probability. It branches out the possible outcomes at each stage, allowing students to follow the paths to the end results—our simple events. By clearly mapping these out, learners can better grasp how the combinations come together and the probability associated with each outcome, enhancing their overall comprehension of basic probability concepts.