Question

A bag contains 4 black and 5 blue marbles. A marble is drawn and then replaced, after which a second marble is drawn. What is the probability that the first is black and second blue?

Short answer

The probability that the first marble is black and the second marble is blue, with replacement, is \(\frac{20}{81}\).

Step-by-step solution

Total Marbles

Calculate the total number of marbles in the bag. There are 4 black marbles and 5 blue marbles, making a total of 9 marbles.

Probability of Picking a Black Marble

Determine the probability of picking a black marble first. There are 4 black marbles in the bag and a total of 9 marbles. Therefore, the probability of picking a black marble first is \(\frac{4}{9}\).

Probability of Picking a Blue Marble

Determine the probability of picking a blue marble next. There are 5 blue marbles in the bag and, since the first marble was replaced, a total of 9 marbles. Therefore, the probability of picking a blue marble after replacing the black marble is \(\frac{5}{9}\).

Calculate the Overall Probability

Multiply the individual probabilities of picking a black marble first (\(\frac{4}{9}\)) and a blue marble next (\(\frac{5}{9}\)) to find the overall probability of both events occurring. The overall probability is \(\frac{4}{9}\) * \(\frac{5}{9}\) = \(\frac{20}{81}\). The probability that the first marble is black and the second marble is blue is \(\frac{20}{81}\).

Key concepts

Probability Theory

Probability theory is a branch of mathematics concerned with the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either occur or not happen at all.

In our context, probability theory involves predicting the likelihood of drawing a specific marble from a bag under certain conditions. To calculate this likelihood, we divide the number of favorable outcomes by the number of all possible outcomes. For instance, if a bag has 4 black and 5 blue marbles, the probability of drawing a black marble first is the number of black marbles (4) divided by the total number of marbles (9), resulting in a probability of \(\frac{4}{9}\).

This exercise requires an understanding of how to calculate these probabilities and then combine them to find the probability of multiple events occurring in sequence, such as drawing two specific marbles one after another.

Independent Events

Independent events are a fundamental concept in probability theory in which the outcome of one event does not affect the outcome of another. For example, in the case of drawing marbles from a bag with replacement, each draw is considered an independent event since the composition of the bag is the same for each draw.

The probability of two independent events occurring in sequence is the product of their individual probabilities. Following our marble example, the probability of first drawing a black marble (\(\frac{4}{9}\)) and then a blue marble (\(\frac{5}{9}\)) is calculated by multiplying these two probabilities, giving us \(\frac{4}{9}\) * \(\frac{5}{9}\) = \(\frac{20}{81}\).

Understanding independence is crucial because it allows us to compute complex probabilities by breaking them down into simpler, individual occurrences.

Combinatorics

Combinatorics is the area of mathematics focused on counting, combination, and permutation of sets of elements. It plays a significant role in probability theory when we deal with discrete events and need to figure out how many ways certain outcomes can occur.

In probability problems like the one presented, combinatorics helps us determine the number of favorable outcomes, which is essential for calculating probabilities. In our case, however, since there is no concern with the order of the outcome nor with combinations of outcomes, the combinatorial aspect is straightforward: we are only interested in calculating the probability of single events and combining them.

Nevertheless, having a grasp on combinatorics can greatly ease the understanding of more complex probability scenarios where the arrangement or grouping of items matters significantly.