Question

Find the number of possible outcomes in the sample space. Then list the possible outcomes. You flip a coin and draw a marble at random from a bag containing two purple marbles and one white marble.

Short answer

The total number of possible outcomes in the sample space is 6, and the possible outcomes are (H, P1), (H, P2), (H, W), (T, P1), (T, P2), (T, W).

Step-by-step solution

Identify the number of outcomes for each task

In the coin flip, there are two possible outcomes: Heads (H) or Tails (T). In drawing a marble, there are three possible outcomes: Purple 1 (P1), Purple 2 (P2), or White (W).

Calculate the total outcomes in sample space

To determine the total outcomes, multiply the outcomes of each task: 2 (coin flip) x 3 (marble draw) = 6 total possible outcomes.

List all the possible outcomes

List all the possible combinations of outcomes: (H, P1), (H, P2), (H, W), (T, P1), (T, P2), (T, W).

Key concepts

Probability

Probability is a way to measure the likelihood of an event happening. In this exercise, we determine the probability of certain events by calculating the possible outcomes from flipping a coin and drawing a marble. To find the probability of a specific outcome, you divide the number of favorable outcomes by the total number of possible outcomes. For example, if you want to know the probability of drawing a white marble, you would count the events where a white marble occurs and divide by 6, since there are 6 possible outcomes overall.

• If a white marble is the only event of interest after a coin flip, it appears twice in the list: once with heads and once with tails, leading to a probability of \( \frac{2}{6} = \frac{1}{3} \).
• For other events, similar calculations help to understand their likelihood.

Understanding probability through everyday tasks like coin flips and marble draws can make statistics more relatable.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting combinations and permutations of sets. It's the backbone of figuring out how many possible outcomes there are in an event, such as our coin and marble problem.In this scenario, you calculate combinations by recognizing each different event (like flipping a coin or drawing a marble) and determining how many choices exist at each step.

• The coin can land in two ways: heads or tails.
• For each flip result, marbles can be chosen in three ways: Purple 1, Purple 2, or White.

By multiplying the number of outcomes from each step, you find the total number of possible outcomes using the basic principle of combinatorics:\[2 \times 3 = 6\]This method allows us to effectively account for every potential sequence of events.

Experimental Outcomes

An experimental outcome is a possible result from a series of actions or experiments. In our example where we flip a coin and draw a marble, each combination is an experimental outcome. Listing all experimental outcomes helps us understand the sample space better. The sample space encompasses every possible combination of results that can occur under the defined actions of flipping a coin and drawing a marble.

• The possible outcomes are combinations of both tasks: (H, P1), (H, P2), (H, W), (T, P1), (T, P2), (T, W).
• Each of these tuples tells us the concrete result of one cycle of the experiment.

Recognizing all these outcomes aids in analyzing probabilities and making predictions about results. Systematically listing combinations ensures no outcomes are overlooked, leading to a more thorough and accurate sample space understanding.