Problem 24 - A First Course in Probability

Question

Each of 2 balls is painted either black or gold and then placed in an urn. Suppose that each ball is colored black with probability \(\frac{1}{2}\) and that these events are independent. (a) Suppose that you obtain information that the gold paint has been used (and thus at least one of the balls is painted gold). Compute the conditional probability that both balls are painted gold. (b) Suppose now that the urn tips over and 1 ball falls out. It is painted gold. What is the probability that both balls are gold in this case? Explain.

Step-by-Step Solution

Answer

The conditional probability that both balls are painted gold given that at least one of the balls is gold is \(\frac{1}{3}\). The conditional probability that both balls are painted gold given that one particular ball is gold is \(\frac{1}{2}\).

Step 1: (1) Ball Colors Possibilities

There are 4 possible outcomes for the 2 balls: (Gold, Gold), (Gold, Black), (Black, Gold), and (Black, Black). 2. Calculate Probabilities of each Outcome: Since there is a probability of \(\frac{1}{2}\) for each color independently, we can calculate the probabilities for each of the four outcomes by multiplying their color probabilities.

Step 2: (2) Outcome Probabilities Calculation

\ \(P(Gold,Gold) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), \(P(Gold,Black) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), \(P(Black,Gold) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), and \(P(Black,Black) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\). a) Compute the conditional probability of both balls being gold given that at least one ball is gold: 3. Calculate Probability of At Least One Ball Gold: To find this conditional probability, we will first calculate the probability of the given information, which is at least one ball being gold.

Step 3: (3) Probability of At Least One Ball Gold

Since the only case that doesn't have at least one gold ball is (Black, Black), we can subtract the probability of (Black, Black) from 1: \(P(\text{At least one Gold}) = 1 - P(Black,Black) = 1 - \frac{1}{4} = \frac{3}{4}\). 4. Calculate Conditional Probability: Now, we can apply the formula for conditional probability: \(P(\text{Both gold}|\text{At least one Gold})=\frac{P(\text{Both gold} \cap \text{At least one Gold})}{P(\text{At least one Gold})}\) Notice that the event (Both gold) is included in (At least one Gold), so their intersection is just (Both gold).

Step 4: (4) Conditional Probability Calculation

\ \(P(\text{Both gold}|\text{At least one Gold})=\frac{P(\text{Both gold})}{P(\text{At least one Gold})}=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}\) Answer for (a): The conditional probability that both balls are painted gold given that at least one of the balls is gold is \(\frac{1}{3}\). b) Compute the conditional probability of both balls being gold given that one particular ball is gold: 5. Use Given Information: The given information for part (b) states that one particular ball is gold, without specifying if it's the first or second ball. We'll assume it is Ball 1.

Step 5: (5) Using Given Information

We know that Ball 1 is gold, so Ball 2 can be either gold or black. In this case, there are only two possibilities: (Gold, Gold) and (Gold, Black). 6. Calculate Conditional Probability: Now, we want to calculate the probability of both balls being gold given that Ball 1 is gold. This can be calculated by finding the probability of (Gold, Gold) in the two scenarios:

Step 6: (6) Conditional Probability Calculation

\ \(P(\text{Both Gold}|\text{Ball 1 is Gold})=\frac{P(Gold,Gold)}{P(Gold,Gold)+P(Gold,Black)} = \frac{\frac{1}{4}}{\frac{1}{4}+\frac{1}{4}} = \frac{1}{2}\) Answer for (b): The conditional probability that both balls are painted gold given that one particular ball is gold is \(\frac{1}{2}\).

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, events, and their likelihood of occurrence, quantified through probabilities.

When we talk about the probability of an event, we are usually referring to the chance that a particular outcome will occur. This is represented by a number between 0 and 1, where 0 indicates an impossible event, and 1 represents a certainty. The probabilities of all possible outcomes in a particular scenario will always add up to 1.

In the case of the urn with gold and black balls, probability theory is applied to determine the likelihood of different combinations of ball colors. The independence of the events, where the color of one ball doesn’t influence the color of the other, allows for straightforward calculations using the basic principles of probability. This simple example serves as an introduction into the more complex rules and applications of probability theory such as conditional probabilities and Bayes' theorem, which are essential tools for making predictions based on incomplete information.

Independent Events

In probability theory, events are considered independent if the occurrence of one does not affect the probability of the other. This is a crucial concept because it simplifies the calculation of probabilities for multiple events.

With independent events, the probability of both events occurring is the product of their individual probabilities. For example, if we were to flip two separate coins, the probability of getting heads on both would be \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\), since the result of one flip does not change the likelihood of the results in the other flip.

In our exercise, the two balls' colors are independent events, which is why we can multiply their individual probabilities to find the probability of both being gold. These probabilities provide the foundation upon which we build when solving more complex problems involving conditional probability, such as determining the likelihood of both balls being gold given that we have some extra information about their colors.

Bayes' Theorem

Bayes' theorem is a powerful tool in probability theory that enables us to update our beliefs about the likelihood of an event based on new evidence. It's named after Thomas Bayes, an 18th-century mathematician and Presbyterian minister who formulated the principle.

This theorem deals with the concept of conditional probability, which is the probability of an event occurring given that another event has already occurred. The theorem’s formula is given by \( P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \), where \( P(A|B) \) is the probability of event \( A \) occurring given that \( B \) has occurred.

In simpler terms, it allows us to revise the chances of event \( A \) happening when event \( B \) is known to have taken place. In the context of our exercise with the golden balls, we use Bayes' theorem to calculate the updated probability of both balls being gold after observing that one or more of the balls is indeed gold. This forms the basis of the conditional probability that we compute in the two parts of the problem, greatly enriching our understanding and approach to tackling real-world problems where prior information plays a key role in decision making.