Bayes Theorem: coin flips and posterior predictive distribution

AI Thread Summary
The discussion focuses on applying Bayes' Theorem to a problem involving three coins with different probabilities of landing heads. The user correctly calculates the posterior probabilities of each coin after observing a head, concluding that the probabilities are 0.1, 0.3, and 0.6 for coins A, B, and C, respectively. They then compute the probability of getting heads again after the first flip, arriving at a final result of 46%. The conversation highlights the utility of probability trees for visualizing complex problems and emphasizes the efficiency of this method compared to direct application of Bayes' Theorem. Overall, the approach demonstrates the flexibility of probability concepts in problem-solving.
Master1022
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Homework Statement
A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the last has 60% chance of coming up heads. I randomly draw a coin from the bag and flip it, and the result comes up heads. What is the probability that if I reflip this coin, it will come up heads again? Why?
Relevant Equations
Bayes Theorem
Hi,

I was attempting the following question and just wanted to check whether my working was correct:

Question: A bag has three coins in it which are visually indistinguishable, but when flipped, one coin has a 10% chance of coming up heads, another as a 30% chance of coming up heads, and the last has 60% chance of coming up heads. I randomly draw a coin from the bag and flip it, and the result comes up heads. What is the probability that if I reflip this coin, it will come up heads again? Why?

Attempt:
Let us denote the coins A, B, and C with probabilities of getting a head as follows: ##p(H|A) = 0.1##, ##p(H|B) = 0.3##, and ##p(H|C) = 0##. Each coin is equally likely to be chosen: ##p(A) = p(B) = p(C) = \frac{1}{3} ##.

Thus, we can find:
$$ p(A|H) = \frac{p(H|A) p(A)}{p(H|A) p(A) + p(H|B) p(B) + p(H|C) p(C)} = \frac{0.1 \cdot \frac{1}{3}}{0.1 \cdot \frac{1}{3} + 0.3 \cdot \frac{1}{3} + 0.6 \cdot \frac{1}{3}} = 0.1 $$
$$ p(B|H) = \frac{p(H|B) p(B)}{p(H|A) p(A) + p(H|B) p(B) + p(H|C) p(C)} = \frac{0.3 \cdot \frac{1}{3}}{0.1 \cdot \frac{1}{3} + 0.3 \cdot \frac{1}{3} + 0.6 \cdot \frac{1}{3}} = 0.3 $$
$$ p(C|H) = \frac{p(H|C) p(C)}{p(H|A) p(A) + p(H|B) p(B) + p(H|C) p(C)} = \frac{0.6 \cdot \frac{1}{3}}{0.1 \cdot \frac{1}{3} + 0.3 \cdot \frac{1}{3} + 0.6 \cdot \frac{1}{3}} = 0.6 $$

So now we can find posterior probability:
$$ p(H|H) = p(H|A)p(A|H) + p(H|B)p(B|H) + p(H|C)p(C|H) = (0.1)^2 + (0.3)^2 + (0.6)^2 = 0.46 $$

Does this look correct?
 
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Looks good to me.
 
I know I always have alternatives, but using a probability tree is a really simple and neat way of doing these problems. Especially if they get more complicated, with more than one factor. In this case we would have:

C1 (1/3) - C1 Heads (1/3 * 1/10 = 1/30) - C1 Heads-Heads (1/300)

C2 (1/3) - C2 Heads (1/3 * 3/10 = 1/10) - C2 Heads-Heads (3/100)

C3 (1/3) - C3 Heads (1/3 * 6/10 = 1/5) - C3 Heads-Heads (12/100)

Heads on first toss (total) = 1/30 + 1/10 + 1/5 = 1/3; Heads on second toss = 46/300

Probability two-heads given first toss is a head = (46/300)/(1/3) = 46/100.
 
PeroK said:
I know I always have alternatives, but using a probability tree is a really simple and neat way of doing these problems. Especially if they get more complicated, with more than one factor. In this case we would have:

C1 (1/3) - C1 Heads (1/3 * 1/10 = 1/30) - C1 Heads-Heads (1/300)

C2 (1/3) - C2 Heads (1/3 * 3/10 = 1/10) - C2 Heads-Heads (3/100)

C3 (1/3) - C3 Heads (1/3 * 6/10 = 1/5) - C3 Heads-Heads (12/100)

Heads on first toss (total) = 1/30 + 1/10 + 1/5 = 1/3; Heads on second toss = 46/300

Probability two-heads given first toss is a head = (46/300)/(1/3) = 46/100.
Thanks @PeroK ! Always great to see other ways to approach a problem; I tend to overcomplicate things. As they say, all roads lead to Rome (however, I guess there are some wrong roads in probability :) ).
 
Master1022 said:
Thanks @PeroK ! Always great to see other ways to approach a problem; I tend to overcomplicate things. As they say, all roads lead to Rome (however, I guess there are some wrong roads in probability :) ).
Just to illustrate the point. Suppose we change the problem so that the three coins are in the proportions ##a_1, a_2, a_3##. You can see immediately how to update the tree structure - and it would only take a few seconds to get the new answer. Whereas, with the raw Bayes Theorem you'd have a bit of work to do.
 
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