Problem 38 - A First Course in Probability

Question

Urn \(A\) has 5 white and 7 black balls. Urn \(B\) has 3 white and 12 black balls. We flip a fair coin. If the outcome is heads, then a ball from urn \(A\) is selected, whereas if the outcome is tails, then a ball from urn \(B\) is selected. Suppose that a white ball is selected. What is the probability that the coin landed tails?

Step-by-Step Solution

Answer

The probability that the coin landed tails, given that a white ball was selected, is \(\frac{12}{85}\).

Step 1: Probability of selecting a white ball from each urn

Let's first find the probability of selecting a white ball from each urn. We are given the number of white and black balls in each urn: Urn A: 5 white balls, 7 black balls (total 12 balls) Urn B: 3 white balls, 12 black balls (total 15 balls) The probability of selecting a white ball from each urn is: Urn A: \(P(W_A) = \frac{5}{12}\) Urn B: \(P(W_B) = \frac{3}{15} = \frac{1}{5}\)

Step 2: Probability of flipping a fair coin

The probability of flipping a fair coin to get heads (selecting a ball from urn A) or tails (selecting a ball from urn B) is equal: \(P(Heads) = P(A) = \frac{1}{2}\) \(P(Tails) = P(B) = \frac{1}{2}\)

Step 3: Use Bayes' theorem

Now that we have the probabilities of selecting a white ball from each urn and the probabilities of flipping a fair coin, we can use Bayes' theorem to calculate the probability of the coin landing tails given that a white ball was selected. Bayes’ theorem states: \(P(B|W) = \frac{P(W|B) P(B)}{P(W)}\) where \(P(B|W)\) = probability of the coin landing tails given that a white ball was selected, \(P(W|B)\) = probability of selecting a white ball from urn B, \(P(B)\) = probability of the coin landing tails, \(P(W)\) = probability of selecting a white ball from any urn.

Step 4: Calculate the probability of selecting a white ball from any urn

To calculate the probability of selecting a white ball from any urn, we can use the law of total probability: \(P(W) = P(W_A)P(A) + P(W_B)P(B)\) Plugging in the values we found in steps 1 and 2, this becomes: \(P(W) = \frac{5}{12} \times \frac{1}{2} + \frac{1}{5} \times \frac{1}{2} = \frac{5}{24} + \frac{1}{10}\) To add the fractions, we need to find a common denominator: \(P(W) = \frac{5}{24} + \frac{12}{24} = \frac{17}{24}\)

Step 5: Apply Bayes' theorem

Now we have all the values needed to apply Bayes' theorem: \(P(B|W) = \frac{P(W_B)P(B)}{P(W)} = \frac{(\frac{1}{5})(\frac{1}{2})}{\frac{17}{24}}\) To find the value of this fraction, we can first multiply the numerators together and then divide by the denominator: \(P(B|W) = \frac{\frac{1}{10}}{\frac{17}{24}}\) Now we can multiply by the reciprocal of the denominator: \(P(B|W) = \frac{1}{10} \times \frac{24}{17} = \frac{24}{170}\) Finally, we can simplify this fraction: \(P(B|W) = \frac{12}{85}\) So the probability that the coin landed tails (selecting a white ball from urn B) given that a white ball was selected is \(\frac{12}{85}\).

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring given that another event has already happened.

Imagine you're watching a soccer game and a player steps up to take a penalty kick. The probability of the player scoring might change if you knew he had successfully made the last five penalty kicks. This is an example of conditional probability, expressed in mathematical terms as P(A|B), meaning the probability of A occurring given that B has taken place.

In our urn and coin problem, conditional probability comes into play as we seek to determine the probability of the coin landing on tails (event B), given that a white ball has been selected (event A). This is written as P(B|A), and navigating this concept is key to applying Bayes' theorem effectively.

Law of Total Probability

The law of total probability is a rule that provides a way to break down complex probability questions into simpler parts. It's like a recipe for a multi-layer cake, where each layer’s ingredients are considered separately before combining them into the final product.

Formally, it's the probability of an event based on several distinct paths or scenarios that could lead to the event. For example, if you have a bag of various colored balls, and you want to calculate the probability of drawing a red ball, you might have several individual bags within the main bag, each with its own mix of red and other colored balls. The law of total probability allows you to consider each sub-bag's contribution to the overall outcome.

In our exercise, this concept comes to life as we calculate the probability of pulling a white ball from any urn without initially specifying from which urn.

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It's the language we use to talk about the uncertainty and predictability of occurrences in the real world, like the rolling of dice, fluctuations in stock prices, or even predictions of weather.

Key to this theory is understanding that probability values range from 0 to 1, where 0 means an event is impossible and 1 means it's certain to happen. The beauty of probability theory lies in its application across varied fields, be it statistics, finance, science, or even philosophy.

The concepts of conditional probability and the law of total probability that we are discussing are fundamental tools within this larger framework of probability theory.

Fair Coin Probability

The term ‘fair coin’ might bring to mind a perfectly balanced coin that, when flipped, has an equal chance of landing on either side—heads or tails. With a fair coin, the probability is split right down the middle: 50% for heads and 50% for tails, or in mathematical terms, P(Heads) = P(Tails) = 0.5.

In probability theory, a coin with an equal chance for heads or tails is an example of a random experiment with two equally likely outcomes. It's a staple example because of its simplicity and symmetry.

This concept plays a significant role in our exercise when we determine the initial chances of selecting from Urn A or Urn B based on a coin flip. The fairness of the coin ensures that no bias affects the probability of where the ball is selected from.