Question

Find the number of possible outcomes in the sample space. Then list the possible outcomes. You draw two marbles without replacement from a bag containing three green marbles and four black marbles.

Short answer

The total number of possible outcomes when drawing 2 marbles without replacement is 21; Specific outcomes can be: (Green-Green), (Green-Black), (Black-Green), (Black-Black)

Step-by-step solution

Identify Total Marbles in Bag

Initially, count the total number of marbles in the bag, which are 7 (3 green + 4 black).

Calculate Number of Possible Outcomes

To calculate the number of ways to draw 2 marbles without replacement from 7, use the combination formula which is C(n, r) = n! / (r!(n-r)!). Here, n represents the total number of items, and r represents the number of items to choose. So, in this case 'n' is 7 (total marbles) and 'r' is 2 (number of marbles to be drawn). Thus, the total number of ways will be C(7,2) = 7! / (2!(7-2)!) = 21.

List Possible Outcomes

Possible outcomes for this experiment when drawing 2 marbles without replacement from the bag can be: Green-Green (GG), Green-Black (GB), Black-Green (BG), and Black-Black (BB). These are four different outcomes to consider. Note, GB and BG are different because once a marble is drawn, it is not replaced.

Key concepts

Outcomes in Sample Space

When tackling problems in probability, it's crucial to first understand what the sample space is. The sample space is the set of all possible outcomes that could occur in an experiment or process. In the case of drawing marbles from a bag, each unique combination of marbles drawn represents an outcome. For example, if you take out two marbles without replacement, the order matters since each draw changes the composition of the bag, resulting in a different outcome.

In our example with seven marbles—three green and four black—the number of outcomes may not seem apparent at first glance, but by considering each possible pair that can be drawn, you'd come up with the different combinations that form the sample space. This includes all the potential combinations of green and black marbles, considering that once a marble is taken out, it's not put back, hence the inventory of marbles changes with each draw.

Combinations Without Replacement

Combinatorics is a field of mathematics that deals with counting, both in a straightforward and abstract manner. In many real-life scenarios, you'll encounter situations where you need to calculate 'combinations without replacement', just like drawing marbles from a bag. This term describes the process of selecting items from a larger set where each selection changes the possible outcomes for the next selection—since the item isn't replaced.

In our marble example, determining the number of combinations involves using a combination formula, since the order of selection doesn't matter and we aren't replacing the drawn marbles. This type of counting problem is essential to understand because it simplifies the process of calculating probabilities in scenarios where you have a limited number of items and selections are made without replacement. It also showcases how each choice affects the remaining options, an important concept in fields like operations research and inventory management.

Factorial Notation

Factorial notation is a mathematical concept that's vital when calculating permutations and combinations. The symbol for a factorial is '!', and it represents the product of all positive integers up to a given number. For instance, the factorial of 5 (written as 5!) is calculated by multiplying 5 by every positive integer below it: 5 x 4 x 3 x 2 x 1, which equals 120.

The concept of factorial is integral to the field of combinatorics because it's used in formulas that find the number of permutations or combinations in a given problem. Permutations refer to the number of ways to arrange items where order does matter, whereas combinations are about grouping items where the order doesn't matter. As shown in the marble problem, we calculated the combinations using the factorial function to determine how many different groups of two marbles can be drawn from the seven available, leading us to the formula involving factorials.