Question

Two events are independent if the probability that one event will occur is not affected by whether or not the other event occurs. For independent events \(\mathrm{A}\) and \(\mathrm{B}\), the probability that A and B will occur equals the probability of A times the probability of B. For example, if you draw a marble from the jar at the right, put it back, and then draw another one, the probability that both marbles are red is \(\frac{3}{5} \cdot \frac{3}{5}=\frac{9}{25}\). A bag contains \(n\) marbles. There are \(r\) blue marbles and the rest of the marbles are yellow. Find the probability of drawing a yellow marble followed by a blue marble if the first one is put back before drawing again.

Short answer

The probability of drawing a yellow marble followed by a blue marble, with replacement, from a bag containing n marbles (r of which are blue), is given by the expression [(n - r) / n] x [r / n].

Step-by-step solution

Calculate the Probability of Drawing a Yellow Marble

Calculate the number of yellow marbles in the bag. Since we know there are r blue marbles, and n total marbles, the number of yellow marbles, y, is given by the equation y = n - r. The probability of drawing a yellow marble (event A), P(A), is then given by the ratio of the number of yellow marbles to the total number of marbles, or P(A) = y/n = (n - r) / n.

Calculate the Probability of Drawing a Blue Marble

The second event (B) is drawing a blue marble. There are r blue marbles. The total number of marbles in the bag remains n. So, the probability for event B, P(B), is P(B) = r / n.

Calculate the Overall Probability

The final step is to find the probability of both events occurring. Given that these are independent events; this will be the product of the probabilities calculated in the previous steps. That is, P(A and B) = P(A) x P(B) = [(n - r) / n] x [r / n].

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events happening. It's the foundation for statistical analysis and guides us in predicting the outcomes of random events. The theory operates on a scale from 0 to 1, where 0 signifies impossibility and 1 indicates certainty.

When we calculate the probability of independent events, like drawing marbles from a bag, we assume that the outcome of one event does not influence the outcome of another. For independent events A and B, the rule for calculating the combined probability of both events occurring is simple: multiply the probability of A by the probability of B. This rule reflects the fact that the odds of A occurring are the same regardless of whether B occurs, and vice versa.

Following the rule for independent events is imperative. To illustrate, if the probability of event A (drawing a yellow marble) is \(P(A) = \frac{(n - r)}{n}\) and the probability of event B (drawing a blue marble) is \(P(B) = \frac{r}{n}\), then the probability of both A and B occurring is \(P(A \text{ and } B) = P(A) \times P(B)\), simply the product of their individual probabilities.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It plays a crucial role in probability theory when determining the number of ways certain events can occur. Combinatoric principles can make calculating probabilities more manageable and more precise.

In the context of our marble example, we are not just interested in counting marbles, but in counting the ways in which they can be drawn. By recognizing the total number of marbles stays constant and each marble has an equal chance to be drawn, we apply combinatoric thinking to calculate probabilities for each drawing. After drawing the first marble and putting it back, the combinatorial situation resets, reminding us that we are dealing with independent events, where the outcome of one draw does not impact the other.

Understanding how to calculate combinations and permutations is critical in solving more complex probability problems, where the order of outcomes matters or where we have without-replacement scenarios. This task, however, hinges on the simpler concept that combinatorics helps us understand each drawing is an independent event with its own probability.

Mathematical Statistics

Mathematical statistics takes the principles of probability theory and applies them to data to make inferences about a population. It uses a combination of probability theory and combinatorics to describe, analyze, and predict real-world phenomena.

In our marble problem, although we are dealing with a straightforward exercise, the methodology touches upon the fundamental concepts of mathematical statistics. Once we have the probabilities for drawing a yellow and a blue marble, we dive into statistical territory. We infer about the likelihood of events (e.g., the drawing of marbles) from the broader context (the entire bag of marbles) by utilizing probability models.

If we were to use this problem in a real-world scenario, we could apply the computed probabilities to make predictions. For instance, we can predict how often a certain color marble would be drawn from the bag over many repeated trials. Understanding this application of statistics is crucial for students who will later encounter more complex probability distributions and hypothesis testing.