1 pointA person predicts daily whether the price of stocks of wrist watch companies will go up or down. If his prediction on stock price of Titan is correct 2 times out of 4, for Rolex it is correct 6 times out of 8 and for Fossil it is correct 5 times out of 9, then what is the probability that at least two of his predictions are correct on a given day?
Question
1 pointA person predicts daily whether the price of stocks of wrist watch companies will go up or down. If his prediction on stock price of Titan is correct 2 times out of 4, for Rolex it is correct 6 times out of 8 and for Fossil it is correct 5 times out of 9, then what is the probability that at least two of his predictions are correct on a given day?
Solution
This problem involves calculating the probability of independent events. Here are the steps to solve it:
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First, calculate the probability of the person predicting correctly for each company:
- For Titan, the probability (P(Titan)) is 2 out of 4, or 0.5.
- For Rolex, the probability (P(Rolex)) is 6 out of 8, or 0.75.
- For Fossil, the probability (P(Fossil)) is 5 out of 9, or approximately 0.56.
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Next, calculate the probability of the person predicting correctly for at least two companies. This can occur in three ways:
- The person predicts correctly for Titan and Rolex, but not Fossil.
- The person predicts correctly for Titan and Fossil, but not Rolex.
- The person predicts correctly for Rolex and Fossil, but not Titan.
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Calculate the probability for each of these scenarios:
- P(Titan and Rolex) = P(Titan) * P(Rolex) * (1 - P(Fossil)) = 0.5 * 0.75 * (1 - 0.56) = 0.165.
- P(Titan and Fossil) = P(Titan) * P(Fossil) * (1 - P(Rolex)) = 0.5 * 0.56 * (1 - 0.75) = 0.07.
- P(Rolex and Fossil) = P(Rolex) * P(Fossil) * (1 - P(Titan)) = 0.75 * 0.56 * (1 - 0.5) = 0.21.
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Add these probabilities together to find the total probability of the person predicting correctly for at least two companies:
- P(at least two correct) = P(Titan and Rolex) + P(Titan and Fossil) + P(Rolex and Fossil) = 0.165 + 0.07 + 0.21 = 0.445.
So, the probability that at least two of his predictions are correct on a given day is approximately 0.445, or 44.5%.
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