Problem 46 - Algebra for College Students

Question

A bag contains five red and 12 white marbles. Two marbles are drawn in succession without replacement. Find the probability of each of the following events:Both marbles drawn are white. \(\frac{33}{68}\)

Step-by-Step Solution

Answer

The probability that both marbles drawn are white is \(\frac{33}{68}\).

Step 1: Identify Total Outcomes

First, calculate the total number of ways to choose any two marbles from the 17 marbles (5 red and 12 white) in the bag. This is calculated using the combination formula \[ \binom{n}{k} = \frac{n!}{k! (n-k)!} \] where \( n \) is the total number of marbles and \( k \) is the number of marbles to pick. Thus,\[ \binom{17}{2} = \frac{17 \times 16}{2 \times 1} = 136 \] So, there are 136 total possible outcomes when drawing two marbles.

Step 2: Calculate Favorable Outcomes

Next, calculate the number of ways to choose two white marbles from the 12 available. This is also a combination problem:\[ \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66 \] Therefore, there are 66 favorable outcomes where both marbles drawn are white.

Step 3: Compute Probability

Finally, compute the probability by dividing the number of favorable outcomes by the total number of possible outcomes:\[ P(\text{Both white}) = \frac{66}{136} = \frac{33}{68} \]The probability that both marbles drawn are white is \(\frac{33}{68}\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and combining objects. It plays a crucial role in solving problems related to probability. In our marble problem, combinatorics helps us determine how many ways we can draw two marbles from a bag.

• Combinatorics involves concepts like permutations and combinations.
• Permutations are about arranging objects, where order matters.
• Combinations focus on selection, where order doesn't matter.

In our problem, we use combinations because we're interested in the selection of marbles without caring about the order. Combinatorial methods give us structured ways to solve complex counting problems, making it easier to tackle real-world scenarios, like card games or even genetic code arrangements.

Probability without Replacement

Probability without replacement refers to scenarios where items are not returned to an original pool after being selected. This concept means that each successive event can affect the probability of the next event.

• In our marble problem, once a marble is drawn, it is not replaced, altering the probabilities.
• This is different from probability with replacement, where drawn items are returned, keeping probabilities consistent.

When the first white marble is drawn, the pool of marbles changes, affecting the outcome and calculations for the next draw. This idea is central to many real-life applications like drawing cards from a deck or lottery draws.

Combination Formula

The combination formula is a mathematical tool used to calculate the number of ways to choose items from a group, ignoring the order of selection. It’s represented as:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here:

• \( n \) is the total number of items.
• \( k \) is the number of items to choose.
• The symbol \( ! \) (factorial) signifies the product of all positive integers up to that number.

In the context of our exercise, using the combination formula helps us find out how many ways we can draw 2 white marbles from a total of 12. By simplifying our calculation with this formula, we make it straightforward, discovering that there are 66 ways to draw two white marbles. The formula is essential for problems where the order doesn’t matter, like lottery combinations or group selections in tournaments.