Question

We have a bag of three biased coins a, b, and c with probabilities of coming up headsof 20%, 60%, and 80%, respectively. One coin is drawn randomly from the bag (withequal likelihood of drawing each of the three coins), and then the coin is flipped threetimes to generate the outcomes X1, X2, and X3.I.) draw the bayesian network corresponding to this setup and define the necessary cpts. Ii.) calculate which coin was most likely to have been drawn from the bag if the observed flips come out heads twice and tails once. Justify your answer.

Answer 1

To find the most likely drawn coin from a bag of three biased coins after observing two heads and one tail from three flips, we create a Bayesian network and define the Conditional Probability Tables. Then, using Bayesian inference, the likelihood of the flip sequence for each coin is calculated, allowing us to determine which coin was most likely to have been drawn.

Step-by-step explanation:

The task requires calculating the likelihood of which biased coin, a, b, or c, was drawn from the bag given that the observed flip outcomes are heads twice and tails once. To approach this problem, we apply Bayesian inference.

I. Bayesian Network and Conditional Probability Tables (CPTs)

Create a Bayesian network with node C representing the choice of coin and nodes X1, X2, X3 representing the flip outcomes. Calculate the CPTs considering the bias probabilities of each coin.

II. Most Likely Coin

Using Bayes' theorem:

Calculate the likelihood of the flip sequence (HH, T) for each coin.

Determine prior probabilities (1/3 for each coin).

Compute the posterior probabilities for each coin being drawn.

Identify the highest posterior probability to conclude the most likely drawn coin.

Generally, in a situation with unequal probabilities for different outcomes, the expected long-term results will align more closely with these probabilities, influencing the most likely outcomes.