Question

Two balls are drawn successively without replacement from an urn containing three white and two red balls. Are the outcomes of the first and second draws independent? Are they exchangeable?

Short answer

No, the draws are not independent, but they are exchangeable.

Step-by-step solution

Defining Probability of Events

First, let's determine the probability of drawing different colors of balls from the urn. The probability of drawing a white ball first is \( P(W_1) = \frac{3}{5} \) and for a red ball is \( P(R_1) = \frac{2}{5} \).

Define Conditional Probabilities

Next, we find the conditional probabilities for the second draw, given what was drawn first. If a white ball is drawn first, the probability of drawing a white ball second is \( P(W_2|W_1) = \frac{2}{4} = \frac{1}{2} \). If a red ball is drawn first, \( P(W_2|R_1) = \frac{3}{4} \).

Check Independence

Two events are independent if \( P(W_2|W_1) = P(W_2) \). We calculate \( P(W_2) = \frac{1}{2}(P(W_2|W_1) \cdot P(W_1)) + \frac{1}{2}(P(W_2|R_1) \cdot P(R_1)) = \frac{1}{2} \cdot \frac{3}{5} + \frac{3}{4} \cdot \frac{2}{5} = \frac{6}{20} + \frac{6}{20} = \frac{6}{10} = \frac{3}{5} \). Since \( P(W_2|W_1) = \frac{1}{2} eq P(W_2) = \frac{3}{5} \), the draws are not independent.

Check Exchangeability

Two events are exchangeable if the sequence of outcomes does not affect the probabilities. We calculate the sequence probabilities and confirm if they match: \( P(W_1 \cap W_2) = \frac{3}{5} \cdot \frac{1}{2} = \frac{3}{10} \) and \( P(W_2 \cap W_1) = \frac{3}{5} \cdot \frac{2}{4} = \frac{3}{10} \). Similarly for \( R_1 \cap R_2 \) and \( R_2 \cap R_1 \). All scenarios match, making outcomes exchangeable.

Key concepts

Conditional Probability

Conditional probability is a critical concept in probability theory that describes the likelihood of an event occurring, given that another event has already occurred. In the context of our exercise, we are interested in the probability of drawing a specific color of ball from the urn on the second draw, given the result of the first draw.

To illustrate, consider the probability of drawing a white ball on the second draw if a white ball was drawn first. This is denoted as \( P(W_2|W_1) \) and calculated by dividing the number of successful outcomes by the total number of possible outcomes remaining after the first draw. If a white ball was drawn first, there are now 2 white balls and 2 red balls left, so \( P(W_2|W_1) = \frac{2}{4} = \frac{1}{2}\).

Conditional probability allows for the adjustment of probabilities based on new information and is foundational for determining independence and exchangeability in probability theory.

Independence of Events

Independence of events is a pivotal concept, allowing us to understand whether the occurrence of one event affects the probability of another. Two events are independent if the occurrence of one does not change the probability of the other occurring. Mathematically, events \( A \) and \( B \) are independent if \( P(A|B) = P(A) \).

In the exercise, let's decide if drawing each ball is independent. We calculated that \( P(W_2|W_1) = \frac{1}{2} \), and found \( P(W_2) = \frac{3}{5} \). Since these probabilities differ, the draws are not independent. The outcome of the first draw changes the likelihood of the resulting event in the second draw. Independence often simplifies probability calculations as this relationship implies events do not influence each other.

Exchangeability

Exchangeability refers to a situation in events where the order of occurrence does not affect the joint probability of events. It emphasizes that the events' sequence does not matter, essentially swapping their occurrence without altering the outcome.

In our exercise, we verified that the sequences of draws maintain the same probabilities irrespective of order. For instance, \( P(W_1 \cap W_2) = \frac{3}{10} \) matches \( P(W_2 \cap W_1) = \frac{3}{10} \). This confirms that the sequences of drawing the balls are exchangeable, meaning that the order does not impact the probabilities of different sequences of events happening. This property often simplifies computations in real-life scenarios where outcomes are observed rather than the specific order of events.

Combinatorics

Combinatorics is the mathematical study of counting, arranging, and dealing with the permutations and combinations of sets. In probability, it is invaluable for solving problems regarding how objects can be selected or arranged.

In the urn problem, combining principles of combinatorics enabled the counting of favorable outcomes. When drawing without replacement, each sequence of events must be considered separately. For instance, the combination of selecting first a white ball and then a red ball is different than selecting a red one first followed by a white. Combinatorics helps calculate probabilities through arrangements or selections, often involving factorial calculations for permutations or combinations.

• Understanding how to use combinations and permutations effectively streamlines complex probability problems.
• Awareness of different arrangements helps ensure no possible outcomes are overlooked.

This knowledge provides a foundation for determining varying probabilities across different scenarios.