Solution
To find the probability that you are flipping the unfair coin given that you observed five tails in a row, we can use Bayes' theorem.
Let's define the events:
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A: You are flipping the unfair coin.
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B: You observe five tails in a row.
Given:
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\( P(A) \): Probability of selecting the unfair coin at random (assume 50% as you pick one coin at random).
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\( P(B|A) \): Probability of observing five tails in a row given that you are flipping the unfair coin. Since the unfair coin always lands tails, \( P(B|A) = 1 \).
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\( P(\neg A) \): Probability of selecting the fair coin at random (also 50% as you pick one coin at random).
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\( P(B|\neg A) \): Probability of observing five tails in a row given that you are flipping the fair coin. This is \( \frac{1}{2^5} = \frac{1}{32} \) since the fair coin has a 50% chance of landing tails on each flip.
We want to find \( P(A|B) \), the probability that you are flipping the unfair coin given that you observe five tails in a row.
We can calculate it using Bayes' theorem:
\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B|A) \times P(A) + P(B|\neg A) \times P(\neg A)} \]
\[ P(A|B) = \frac{1 \times 0.5}{1 \times 0.5 + \frac{1}{32} \times 0.5} \]
\[ P(A|B) = \frac{0.5}{0.5 + \frac{1}{64}} \]
\[ P(A|B) = \frac{0.5}{\frac{32}{64} + \frac{1}{64}} \]
\[ P(A|B) = \frac{0.5}{\frac{33}{64}} \]
\[ P(A|B) = \frac{0.5 \times 64}{33} \]
\[ P(A|B) = 0.9697 \]
So, the probability that you are flipping the unfair coin given that you observe five tails in a row is approximately 0.9697, or 96.97%.