Problem 32 - 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.) The marbles are of different colors.

Step-by-Step Solution

Answer

The probability of drawing two marbles of the same color is \(\frac{4}{15}\).

Step 1: Identify the total number of marbles

Count the total marbles present in the bag. One green, two yellow and three red marbles implies there are 6 marbles in total.

Step 2: Calculate sample space

To find the total number of ways two marbles can be drawn from a bag containing 6, use combination notation \(\binom{n}{r}\), also represented as \(C(n, r)\) or 'n choose r'. Here, \(n\) is the total number of items, and \(r\) is the items to choose. Hence, Calculate \(C(6, 2) = \frac{6!}{2!(6-2)!} = 15\). This means the sample space, or total possible outcomes, is 15.

Step 3: Calculate event outcome

The event is drawing two marbles of the same color. For two marbles to be of the same color, they can be either both yellow or both red. Green can't be considered as there is only one green marble. Therefore, calculate the number of ways for these two scenarios to occur: drawing two yellow marbles \(C(2, 2) = 1\) and drawing two red marbles \(C(3, 2) = 3\). Adding them together gives a total of 4 outcomes for the event.

Step 4: Calculate Probability

The probability of an event is the ratio of the desirable outcomes to the total outcomes. Hence, the probability \(P\) is calculated as \(P = \frac{number \, of \, favorable \, outcomes}{total \, outcomes}\), or \(P = \frac{4}{15}\)

Key Concepts

Combinations

In probability and statistics, combinations play a crucial role when determining possible selections of items where the order does not matter. When you hear the term "combination," think of it as a selection of items. In the problem of drawing marbles, we used combinations to find how many ways we can draw two marbles from a total of six. The mathematical notation for combinations is \(C(n, r) = \frac{n!}{r!(n-r)!}\)where:

• \(n\) is the total number of items,
• \(r\) is the number of items to be chosen, and
• \(!\) denotes a factorial, which means multiplying a series of descending natural numbers.

Understanding combinations allows us to calculate the sample space, which reflects all possible outcomes of an event. Here, this concept helped us conclude that there are 15 different combinations of drawing two marbles from six.

Sample Space

Sample space is a fundamental concept in probability, representing all possible outcomes of an experiment. For instance, if you roll a six-sided die, the sample space is \(\{1, 2, 3, 4, 5, 6\}\). It encompasses every possible result that could occur when the experiment is conducted. In the marble drawing exercise, we find the sample space using combinations. We computed \(C(6, 2) = 15\)which tells us there are 15 possible ways to draw two marbles from a total of six.

• This includes all possible pairs regardless of the colors.
• The sample space helps in calculating the likelihood or probability of different events.

By understanding the sample space, you can identify the total number of potential outcomes against which you can measure the specific event outcomes.

Marbles

Marbles in probability problems are often used because they provide a tangible example of random selection. Each marble represents a unique unit, and the attributes (like color) determine different outcomes. In our example, the bag contains:

• 1 green marble,
• 2 yellow marbles, and
• 3 red marbles.

These marbles allow us to explore the concept of drawing without replacement, meaning once you pick a marble, it's not put back into the bag for further draws. It changes the total available counts and can significantly impact the probability of different outcomes. Dealing with marbles and their unique colors demonstrate how combinations and sample spaces work together in probability theory.

Event Outcome

The event outcome in a probability question refers to the actual result we are interested in calculating. In the given exercise, we're focused on the event of drawing two marbles of the same color. To find this, we evaluated each possible scenario:

• Drawing two yellow marbles: \(C(2, 2) = 1\) way
• Drawing two red marbles: \(C(3, 2) = 3\) ways

Together, these give us 4 desirable outcomes.Once the event outcomes are recognized, calculating the probability becomes straightforward. We simply divide the number of favorable outcomes by the total outcomes from the sample space. So, for our example, the probability is \(P = \frac{4}{15}\).Understanding event outcomes allow you to pinpoint exactly what scenarios in your sample space match the event criteria.