Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome-heads or tails. The first coin is a two-headed coin, the second is a biased coin such that \(P(\mathrm{H})=.75\), and the third is a fair coin. a. What is the probability that the coin that is tossed will show heads? b. If the coin selected shows heads, what is the probability that this coin is the fair coin?

Step-by-Step Solution

Answer

a. The probability that the tossed coin will show heads is \(\frac{4}{6}\). b. If the coin selected shows heads, the probability that this coin is the fair coin is \(\frac{1}{4}\).

Step 1: Define the Events and Probabilities

We begin by defining the events: A1: Selecting the two-headed coin A2: Selecting the biased coin A3: Selecting the fair coin H: Tossing a coin and getting heads The probabilities of selecting each coin are equal and can be defined as follows: P(A1) = P(A2) = P(A3) = 1/3

Step 2: Find the Probabilities of Getting Heads

We need to find the probabilities of getting heads given that we selected each coin. P(H|A1) = 1 (As this is a two-headed coin) P(H|A2) = 0.75 (As given in the problem) P(H|A3) = 0.5 (As this is a fair coin)

Step 3: a. Calculate the Total Probability of Getting Heads

To find the total probability of getting heads after tossing the coin, we can use the Law of Total Probability. This law states that the probability of an event (H) can be calculated by summing the probabilities of each possible cause (A1, A2, or A3) and the probability of the event (H) given that cause. P(H) = P(H|A1)P(A1) + P(H|A2)P(A2) + P(H|A3)P(A3) Now, substituting the given probabilities, we get: P(H) = (1)(1/3) + (0.75)(1/3) + (0.5)(1/3) P(H) = 1/3 + 1/4 + 1/6 P(H) = 4/6 #Answer_a#: The probability that the tossed coin will show heads is 4/6.

Step 4: b. Calculate the probability that the coin showing heads is the fair coin

Now we need to find the probability that the coin showing heads is the fair coin. For this, we can use Bayes' theorem. Bayes' theorem states that the conditional probability P(A3|H) can be found using the following formula: P(A3|H) = P(H|A3)P(A3) / P(H) We have already calculated P(H) in step a, and we know P(H|A3) and P(A3) from step 1. Now, we substitute these values into the formula: P(A3|H) = (0.5)(1/3) / (4/6) Simplify the expression: P(A3|H) = (0.5)(1/3) / (2/3) P(A3|H) = 1/4 #Answer_b#: If the coin selected shows heads, the probability that this coin is the fair coin is 1/4.

Key Concepts

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. For instance, if we consider tossing a coin, the probability of getting heads might be different if we know the coin is biased. In the context of the exercise, the conditional probability can be represented as P(H|Ai), where H denotes heads and Ai represents the event of selecting the i-th coin.

Understanding conditional probability is crucial because it allows us to update our beliefs about the likelihood of events based on new information. For example, if you knew you had selected a two-headed coin, you would be absolutely certain of getting heads—this is reflected by a conditional probability of 1. In the textbook exercise, each coin has a different likelihood of showing heads when tossed, and knowing which coin is selected alters the probability of seeing heads on top.

There are many real-world applications for conditional probability, from predicting whether it will rain given the current humidity levels, to something as serious as calculating the likelihood of a medical condition given certain symptoms.

Law of Total Probability

The Law of Total Probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It essentially asserts that if we have a set of mutually exclusive and exhaustive events, the total probability of any event can be found by considering each possible scenario that could lead to that event.

For example, when you flip a coin, there are two possible outcomes: heads or tails. However, if you had multiple coins with varying probabilities of landing on heads, the Law of Total Probability allows you to account for each coin's likelihood and sum these up to find the total probability of getting heads. In the exercise, it is used to combine the probabilities of getting heads from each of the three different coins, considering they are selected with equal chance.

This law is often applied in scenarios with stage-wise processes or classifications, such as determining the probability of a computer part being defective given different manufacturing lines or, as seen in the exercise, calculating the overall probability of a random coin toss resulting in heads.

Bayes' Theorem

Bayes' Theorem is a powerful formula used to find the probability of an event based on prior knowledge of conditions that might be related to the event. Its essence is to revise our predictions given new evidence.

In our coin toss exercise, Bayes' Theorem helps answer a reverse probability question: If we observe heads, what's the chance that the fair coin was tossed? Bayes' Theorem is applied after we witness the outcome (heads in this case) to update our belief about which coin was likely used. With the formula \[ P(A3|H) = \frac{P(H|A3)P(A3)}{P(H)} \], we find the updated probability of the fair coin being the one tossed, conditioned on the fact that heads appeared.

Beyond the textbook, Bayes' Theorem is crucial in various fields like medical diagnosis (estimating the probability of a disease given a test result), machine learning (for updating the model as new data arrives), and even in the judicial system (re-evaluating the likelihood of guilt with each new piece of evidence).