Problem 45 - 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 red. \(\frac{5}{68}\)

Step-by-Step Solution

Answer

The probability of drawing two red marbles is \(\frac{5}{68}\).

Step 1: Determine Total Number of Marbles

First, identify the total number of marbles in the bag. There are 5 red marbles and 12 white marbles. Hence, the total number of marbles is:\[\text{Total Marbles} = 5 + 12 = 17\]

Step 2: Calculate Probability of Drawing the First Red Marble

To find the probability of drawing the first red marble from the total of 17 marbles, we use the formula for probability:\[P(\text{First Red}) = \frac{\text{Number of Red Marbles}}{\text{Total Number of Marbles}} = \frac{5}{17}\]

Step 3: Calculate Probability of Drawing the Second Red Marble

After drawing one red marble, there are 4 red marbles left and a total of 16 marbles left. Therefore, the probability of drawing a second red marble is:\[P(\text{Second Red | First Red}) = \frac{4}{16} = \frac{1}{4}\]

Step 4: Multiply Probabilities for Successive Events

Since the events are successive, we calculate the probability of both events happening by multiplying the probabilities obtained from Steps 2 and 3:\[P(\text{Two Reds}) = P(\text{First Red}) \times P(\text{Second Red | First Red}) = \frac{5}{17} \times \frac{1}{4}\]

Step 5: Simplify the Result

Calculate the multiplication from Step 4 and simplify:\[P(\text{Two Reds}) = \frac{5}{17} \times \frac{1}{4} = \frac{5}{68}\]This results in the final probability of drawing two red marbles in succession without replacement being \(\frac{5}{68}\).

Key Concepts

Conditional Probability

Conditional probability is a concept that refers to finding the likelihood of an event occurring, given that another event has already occurred. It's quite handy in situations like the one described in the marble problem, where sequences of actions are involved.
To calculate conditional probabilities, we usually rely on the formula:

• \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
• The notation \( A|B \) implies the probability of event A occurring given that event B has already happened.

In practice, with our marble problem, the probability of drawing the second red marble (after already drawing one) is calculated as a conditional probability. The pool of marbles changes if we don't replace the first marble drawn. Hence, the calculation depends on the new condition: there are fewer marbles remaining after the first draw. This is where conditional probability is critical, helping us understand probabilities in dynamic circumstances.

Combinatorial Analysis

Combinatorial analysis is crucial when dealing with probability problems, especially when solutions involve counting outcomes precisely. This kind of analysis helps us determine the potential sequences or combinations that could occur in a random process.
Typically, combinatorial methods include techniques like counting the number of ways we can choose items without regard to order (combinations) or with order (permutations).
Let's tie this back to our marble problem:

• We could use combinations to assess the number of distinct ways to select two marbles from the bag. But here, since the question specifies drawing in succession, a straightforward count suffices.
• This direct count assumes no replacement, thus altering the number and type of marbles after each draw. This sequence-driven calculation is why a combinatorial understanding is essential.

By visualizing the situation through combinations, we can comprehend the vast yet specific range of potential outcomes, even before delving into exact probability calculations.

Basic Probability Concepts

Understanding basic probability is foundational for solving any probability-related problem. Probability quantifies the chance or likelihood of an event occurring. It's generally expressed as a fraction, a decimal, or a percentage.
The basic probability formula is:

• \[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

For the marbles, we started by finding the total number of marbles, providing a clear denominator for our probability calculations.

• This sum of red and white marbles gives us the total outcomes.
• The number of favorable outcomes—drawing a red marble—is our numerator.

In our case, calculating these simple probabilities and multiplying them is how we derive the likelihood of drawing two red marbles successively. Understanding these basics lets us tackle more complex scenarios with confidence.