Puzzle Answer #3
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By
Bayes's Theorem
Conditional Probability: 3 possible outcomes are illustrated below: |
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| For the Two-headed Coin: | |||
| Outcome 1: | Pick the Two-headed
coin with probability = 0.5, the probability it shows a head = 1 |
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| For the Normal Coin: | |||
| Outcome 2: | Pick the Normal
coin with probability = 0.5, the probability it shows a head = 0.5. |
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| Outcome 3: | Pick the Normal
coin with probability = 0.5, the probability it shows a tail = 0.5. |
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| The Conditional Probability of an event A, given that an event B has occurred | |||
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= |
P(A | B) | ||
| where Event A = the probability that the other side is a head Event B = given that the occurrence of a head |
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| P(A | B) | |||
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= |
P(AÙB) / P(B) | ||
| = | P(Outcome 1) / [P(Outcome 1) + P(Outcome 2)] | ||
| = | (0.5 * 1) / [(0.5 * 1) + (0.5 * 0.5)] | ||
| = | 0.5 / 0.75 | ||
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= |
½ / ¾ | ||
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= |
⅔ | ||
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| By
Wording Explanation: If you don't know Bayes's Theorem or you think it's a little messy from the calculation above. You may think about this: |
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| The
question is "one
coin is selected from the box at random and the face of one side is
observed as a head, what is the probability that the other side is also a
head?". It means that a coin have already been selected; it shows a head and you want the other side is also a head. It is different from that you want to pick the two-headed coin. Think about this: Since a coin have already been selected and it shows a head, there were 3 heads to begin with. For these 3 heads, 2 of them have another head on the other side. With equal probability of picking one of the 3 heads, you do have 2 out of 3 with a head on the reverse side. Therefore, the answer is ⅔.
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