To solve this problem using Bayes' Theorem, let's define the relevant events and fill in the values for the formula.

Problem Setup:

• $A$: Event that coin A is selected.
• $B$: Event that coin B is selected.
• $H$: Event that the selected coin comes up heads.

Given probabilities:

• $P(A) = 0.4$ (probability of selecting coin A)
• $P(B) = 0.6$ (probability of selecting coin B)
• $P(H|A) = 0.3$ (probability of heads given coin A is selected)
• $P(H|B) = 0.9$ (probability of heads given coin B is selected)

Goal:

We need to find the probability that coin B was chosen given that the result was heads, i.e., $P(B|H)$. Bayes' Theorem can be applied as follows:

$P(B|H) = \frac{P(H|B) \cdot P(B)}{P(H)}$

Where:

• $P(H|B)$: The probability of getting heads given coin B was selected (this is 0.9).
• $P(B)$: The probability of selecting coin B (this is 0.6).
• $P(H)$: The total probability of getting heads.

Now, $P(H)$ (the total probability of getting heads) can be found using the Law of Total Probability:

$P(H) = P(H|A) \cdot P(A) + P(H|B) \cdot P(B)$

Substitute the known values:

$P(H) = (0.3 \cdot 0.4) + (0.9 \cdot 0.6)$ $P(H) = 0.12 + 0.54 = 0.66$

Now apply Bayes' Theorem:

$P(B|H) = \frac{0.9 \cdot 0.6}{0.66}$ $P(B|H) = \frac{0.54}{0.66} \approx 0.818$

Conclusion:

The probability that coin B was chosen given that the result was heads is approximately 0.818.

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In terms of Bayes' Theorem, the values are:

$P(B|H) = \frac{P(H|B) \cdot P(B)}{P(H)}$

• $X = P(H|B) = 0.9$
• $Y = P(B) = 0.6$
• $Z = P(H) = 0.66$

Would you like to discuss this further or go into more details?

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Here are five related questions to deepen your understanding:

1. How would the result change if the probabilities of selecting coins A and B were equal?
2. What if the probability of heads for coin A was higher than for coin B?
3. How can Bayes' Theorem be used in other real-world applications?
4. What is the importance of the Law of Total Probability in this problem?
5. How would the probability of tails change the analysis?

Tip: When applying Bayes' Theorem, ensure you carefully define the events and compute the total probability for better clarity.