Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. (Hint: Use combinations to find the numbers of outcomes for the given event and sample space.) Both marbles are red.

Step-by-Step Solution

Answer

The probability that both marbles drawn are red is 0.2

Step 1: Compute Total Number of Ways to Draw Two Marbles

Since there are total of 6 marbles, the number of ways to draw two marbles from this lot can be calculated using the combination formula: \(\text{C(n, k)} = n! / [(n - k)! k!]\), where n is the total number of items, and k is the number of items to choose. Therefore, the total number of ways to draw two marbles are \(\text{C(6, 2)} = 15\)

Step 2: Compute Number of Ways to Draw Two Red Marbles

The next step is to calculate the number of ways to draw two red marbles from the three available. Using the same combination formula, there is \(\text{C(3, 2)} = 3\) ways to do so.

Step 3: Compute Probability

Probability is defined as the number of desirable outcomes divided by the total number of outcomes. Therefore, the probability that both marbles drawn are red is the number of ways to draw two red marbles divided by the total number of ways to draw two marbles. This probability is \(\text{P(Both Red)} = \text{C(3, 2)} / \text{C(6, 2)} = 3/15 = 0.2\)

Key Concepts

Combinations

When you're dealing with probabilities and choosing items from a set, combinations are your best friends. Combinations help you calculate how many different ways you can select items without considering the order. That's important because, in many experiments and scenarios, the order in which you choose items doesn't change the outcome.

The formula for combinations is: \[ \text{C(n, k)} = \frac{n!}{(n-k)!k!} \]Here:

• \(n\) is the total number of items.
• \(k\) is the number of items you want to select.
• '!' denotes factorial, which means multiplying the number by all the positive integers less than itself.

This formula is handy for solving problems like finding out how many ways you can pick 2 marbles out of 6, just as in the exercise example. The result is a selection of possibilities without concern for selection order.

Experimental Outcomes

In an experiment, each possible result is known as an outcome. These are the ways an experiment can turn out based on various conditions.

Let's consider the experiment of drawing two marbles from a bag filled with different colored marbles. Each pair of marbles drawn is a different outcome.

• One possible outcome might be drawing two red marbles.
• Another could be drawing one green and one yellow marble.
• Each possible pair is an experimental outcome.

The concept of experimental outcomes allows you to quantify and visualize what might happen as the result of an experiment. It helps you list all potential results before moving to the next step – calculating probabilities.

Event Probability

Event probability is all about finding how likely a particular outcome or a group of outcomes, referred to as an event, is to happen in a given experiment.

To find the probability of drawing two red marbles, for instance, you calculate the total number of favorable outcomes and divide it by the total number of possible outcomes.

• Favorable outcomes: drawing two red marbles (3 ways).
• Total possible outcomes: drawing any two marbles from the bag (15 ways).

Hence, the probability of the event "drawing two red marbles" is:\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}} = \frac{3}{15} = 0.2 \]This probability tells us that there's a 20% chance of drawing two red marbles, providing a clear measurement of our chances for a specific event.

Sample Space

The sample space of an experiment includes every possible outcome it can have. For drawing marbles from a bag, it's critical to first design this complete set. This way, you understand what all can happen before narrowing it down to specific events.

• The sample space for drawing 2 marbles from a bag with 6 distinct choices (green, yellow, red) would include all pairing combinations.
• This means combinations like green & yellow, yellow & red, red & red, etc.

Understanding the sample space is vital because it sets the stage for determining probabilities. Knowing all outcomes helps in comparing specific events against all possibilities, giving accurate probability calculations.