Question

If a bag contains three white, two black, and four red balls and four balls are drawn at random with replacement, calculate the probabilities that (a) The sample contains just one white ball. (b) The sample contains just one white ball given that it contains just one red ball.

Short answer

(a) The probability of the sample containing just one white ball is \(\frac{40}{729}\). (b) The probability of the sample containing just one white ball given it contains just one red ball is \(\frac{8}{243}\).

Step-by-step solution

(a) Probability of the sample containing just one white ball

To have just one white ball in the sample of four balls drawn, it could happen in 4 different ways: 1. WWBR (W - White, B - Black, R - Red) 2. WBWR 3. WRWB 4. WRRW We need to find the probability of each of these occurrences happening. 1. WWBR: Probability = \(\frac{1}{3} \times \frac{2}{9} \times \frac{1}{3} \times \frac{4}{9} = \frac{8}{729}\) 2. WBWR: Probability = \(\frac{1}{3} \times \frac{2}{9} \times \frac{4}{9} \times \frac{1}{3} = \frac{8}{729}\) 3. WRWB: Probability = \(\frac{1}{3} \times \frac{4}{9} \times \frac{2}{9} \times \frac{1}{3} = \frac{8}{729}\) 4. WRRW: Probability = \(\frac{1}{3} \times \frac{4}{9} \times \frac{4}{9} \times \frac{1}{3} = \frac{16}{729}\) Now, sum all those probabilities to find the overall probability of having just one white ball: Probability (a) = \(\frac{8}{729} + \frac{8}{729} + \frac{8}{729} + \frac{16}{729} = \frac{40}{729}\)

(b) Probability of the sample containing just one white ball given it contains just one red ball

In this case, we already know that the sample contains just one red ball, so we have three remaining balls to draw, which must be two black balls and one white ball. The sample could be one of the following: 1. WWBR 2. WBWR 3. WRWB Notice that we only have three possibilities since we are given the presence of one red ball. We already calculated their probabilities in part (a): 1. WWBR: Probability = \(\frac{8}{729}\) 2. WBWR: Probability = \(\frac{8}{729}\) 3. WRWB: Probability = \(\frac{8}{729}\) Now, sum all those probabilities to find the overall probability of having just one white ball given it contains just one red ball: Probability (b) = \(\frac{8}{729} + \frac{8}{729} + \frac{8}{729} = \frac{24}{729} = \frac{8}{243}\) So the final answers are: (a) Probability = \(\frac{40}{729}\) (b) Probability = \(\frac{8}{243}\)

Key concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with the study of countable discrete structures. It involves several subfields, such as counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria. In the context of our exercise, combinatorics helps us determine the different possible combinations of balls that can be drawn from the bag.

When you're solving problems like the given example, combinatorics comes into play when identifying all possible combinations of the balls that could be drawn, which is essential before calculating probabilities. Specific to this case, sequences such as 'WWBR' or 'WBWR' illustrate some of these combinations. By establishing the total number of valid combinations and then finding the probability for each one, you get closer to the final solution.

Probability Theory

Probability theory is a subfield of mathematics concerned with analyzing random events. 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 singularly, or evolve over time in an apparently random fashion.

In our exercise, we consider the event of drawing balls from a bag. The probability of drawing a ball of a certain color is the ratio of the number of balls of that color to the total number of balls. Because the drawing is with replacement, the probabilities do not change from draw to draw. Calculating the probability of a sequence of draws, like 'WWBR', involves multiplying the probabilities of each draw, as each draw is independent.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying 'smoothly', the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly, but have distinct, separated values.

Probability calculations involving drawing balls from a bag, as in our exercise, are an example of a problem in discrete mathematics. The balls can only come in whole counts; hence, the mathematics involved is discrete. Additionally, when calculating probabilities, we consider individual outcomes that are distinct and separable events, a perfect example of discrete analysis.

Random Sampling with Replacement

Random sampling with replacement is a technique used to gather a sample from a set where each member of the set can be chosen more than once. In the case of our exercise, once a ball is drawn and its color is noted, it is put back into the bag before the next ball is drawn. This implies that the same ball can potentially be drawn multiple times.

Sampling with replacement is important because it ensures that the probability of drawing each ball remains constant from one draw to another. The calculations shown in the step-by-step solution for (a) and (b) implicitly use this principle. When handling probabilities in this context, it's crucial to apply the rule that the probability of multiple independent events all occurring is the product of their individual probabilities.