Problem 53 - College Algebra with Modeling and Visualization

Question

\(\quad\) A jar contains 22 red marbles, 18 blue marbles, and 10 green marbles. If a marble is drawn from the jar at random, find the probability that the color is the following. A. Red B. Not red C. Blue or green

Step-by-Step Solution

Answer

A. \(\frac{11}{25}\); B. \(\frac{14}{25}\); C. \(\frac{14}{25}\).

Step 1: Calculate Total Number of Marbles

To determine the probability of drawing a marble of a certain color, we first need to know the total number of marbles in the jar. The number of each color is given: 22 red, 18 blue, and 10 green. Sum these quantities to find the total number of marbles: \[22 + 18 + 10 = 50\]So, there are 50 marbles in total.

Step 2: Probability of Drawing a Red Marble

The probability of drawing a red marble is the ratio of the number of red marbles to the total number of marbles. There are 22 red marbles out of 50 marbles in total, so the probability is:\[P( ext{Red}) = \frac{22}{50} = \frac{11}{25}\]

Step 3: Probability of Not Drawing a Red Marble

The probability of not drawing a red marble involves the complement of drawing a red marble. First, calculate how many marbles are not red. In our jar, the total number of non-red marbles is the sum of blue and green marbles, i.e., 18 blue + 10 green = 28. Hence, the probability is:\[P( ext{Not Red}) = \frac{28}{50} = \frac{14}{25}\]

Step 4: Probability of Drawing a Blue or Green Marble

The probability of drawing a blue or green marble is essentially the probability of not drawing a red marble, as the events are complementary. Nevertheless, using the independent events, the total number of blue or green marbles is 18 blue + 10 green = 28. Therefore, the probability is: \[P( ext{Blue or Green}) = \frac{28}{50} = \frac{14}{25}\]

Key Concepts

Complementary Events

In probability, complementary events are pairs of outcomes that do not overlap and together cover all possible scenarios. For example, consider the exercise of drawing marbles from a jar. Here, the complementary events are:

• Drawing a red marble
• Not drawing a red marble (which means drawing either a blue or green marble)

These two scenarios together account for every possible outcome when a marble is drawn. Complementary events always sum to 1 in probability. Mathematically, this is expressed as:\[ P(A) + P(A') = 1 \]where \(A\) is the event of drawing a red marble and \(A'\) is the complementary event of not drawing a red marble. Knowing the probability of one event helps quickly find the probability of its complement by simply subtracting from 1. This concept greatly simplifies problems where you need to find the probability of event(s) not occurring.

Independent Events

Independent events in probability are scenarios where the outcome of one event does not affect the outcome of another. When drawing a marble from a jar, each draw can often be considered independent if the marble is replaced, although in this problem, replacement is not mentioned.However, for some aspects of the exercise, drawing different colors such as blue or green can be treated with separate calculations as though they were independent events in the probability sense because we are merely summing probabilities to find combined outcomes. Consider the red, blue, and green marbles. The probability of drawing either blue or green is analyzed:

• Probability of drawing a blue marble: \( \frac{18}{50} \)
• Probability of drawing a green marble: \( \frac{10}{50} \)

That's verified by summing the outcomes, assuming the draws do not affect each other, leading to efficient probability calculations. While these results stem from complementary event principles, they illustrate how considering events independently can simplify the probability assessment.

Total Probability

The total probability of all possible outcomes in an event must equal 1, reflecting certainty that some outcome will definitely happen. In our exercise, the total probability is enhanced by considering all different colored marbles – red, blue, and green.To ensure calculation accuracy, all individual probabilities must add up to 1, demonstrating that one of the options must occur. This is done by confirming:

• Probability of a red marble: \( \frac{11}{25} \)
• Probability of a non-red marble (blue or green): \( \frac{14}{25} \)

Summing these produces a total probability equation:\[ \frac{11}{25} + \frac{14}{25} = \frac{25}{25} = 1 \]This ensures comprehension of probability through direct and inverse count methods, making the probabilities clear and providing a robust framework to verify the correctness of all calculated outcomes.