Stock Movement Accuracy

Solution & Explanation

To solve the problem of determining the probability that a stock will actually increase given that a software application predicts it will, we need to apply Bayes' Theorem. Bayes' Theorem allows us to update our prior beliefs with new evidence.

Definitions and Assumptions:

• P(A): Probability that the stock actually goes up.
  • Assumed to be 0.5 (50%) based on a fair assumption of equal chance of going up or down.
• P(B|A): Probability that the program predicts the stock will go up, given that it actually goes up.
  • Given as 0.6 (60% accuracy rate).
• P(B): Probability that the program predicts the stock will go up.
  • This can be calculated using the law of total probability.
• P(A|B): Probability that the stock actually goes up given that the program predicts it will go up.
  • This is what we want to calculate.

Calculating P(B):

To find P(B), we use the law of total probability:

$P(B) = P(B|A) \cdot P(A) + P(B|A^C) \cdot P(A^C)$

Where:

• P(A^C): Probability that the stock does not go up (goes down), which is 1 - P(A) = 0.5.
• P(B|A^C): Probability that the program predicts the stock will go up, given that it actually goes down.
  • Since the program's accuracy is 60%, the probability of a wrong prediction is 40%, so P(B|A^C) = 0.4.

Substituting the values:

$P(B) = 0.6 \cdot 0.5 + 0.4 \cdot 0.5 = 0.3 + 0.2 = 0.5$

Applying Bayes' Theorem:

Bayes' Theorem is given by:

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

Substitute the known values:

$P(A|B) = \frac{0.6 \cdot 0.5}{0.5} = \frac{0.3}{0.5} = 0.6$

Conclusion:

The probability that the stock will actually go up given that the program predicts it will, is 0.6 or 60%. This result indicates that the program's prediction aligns with its accuracy, reinforcing the assumption that the stock has an equal chance of going up or down without additional market information.