Problem 43 - Algebra for College Students

Question

A bag contains one red and two white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are white. \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{3}\).

Step 1: Understanding the Problem

We need to find the probability that both marbles drawn from the bag are white. Initially, the bag contains one red marble and two white marbles.

Step 2: Determine Total Number of Outcomes

The total number of ways to draw two marbles from three is calculated using combinations. This is given by: \(\binom{3}{2} = 3 \) outcomes.

Step 3: Identify Favorable Outcomes

The event of both marbles being white can occur if we pick both white marbles. Initially, there are 2 white marbles in the bag, so the number of favorable outcomes is: \(\binom{2}{2} = 1\) outcome.

Step 4: Calculate Probability

The probability is the number of favorable outcomes divided by the total number of outcomes: \(P(\text{both white}) = \frac{1}{3}\).

Key Concepts

Combinations

Combinations are a way to select items from a larger set, where the order of selection does not matter. This concept is central to understanding probability, especially when it comes to events involving multiple selections, like drawing marbles from a bag.
When calculating combinations, we use the formula for binomial coefficients: \ \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \) \ Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items to be chosen. The factorial symbol (!), represents the product of an integer and all the integers below it. For instance, the factorial of 3 (\(3!\)) is equal to \(3 \times 2 \times 1 = 6\).
In the exercise, we calculated the total number of ways to draw two marbles from three using this formula, resulting in three possible selections. The calculation was:

• \( \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \)

This indicates there are three possible ways to choose two marbles out of the three available.

Favorable Outcomes

Favorable outcomes refer to the specific outcomes that align with the event of interest—in this case, drawing two white marbles from the bag. It is an important part of determining probability because it details the scenarios we are interested in measuring.
In the context of our exercise, a favorable outcome is successfully drawing both white marbles. Since there are two white marbles and we are selecting both, the number of favorable outcomes is calculated in a similar manner to combinations as follows:

• \( \binom{2}{2} = 1 \)

This calculation indicates that there is only one way to pick both white marbles from the two available. By identifying this, we can understand that there are very limited scenarios where our specific event (both marbles being white) occurs, making it crucial in calculating our final probability.

Total Outcomes

Total outcomes are the number of all potential results in a probability experiment. This includes every possible combination that could occur given the situation. Knowing the total number of outcomes allows us to assess the probability of each individual outcome.
In the example provided, we found that the total number of ways to draw two marbles from a set of three is three. This is essential because it sets the denominator of our probability fraction:

• The calculation for total outcomes was \( \binom{3}{2} = 3 \).

By establishing this measure, it becomes easier to compare it against the number of favorable outcomes. In the end, probability is determined by the ratio of favorable outcomes to total outcomes. For the event of drawing both white marbles, it was calculated as \( P(\text{both white}) = \frac{1}{3} \), highlighting how relative rarity or commonality of an event is measured within a given scenario.