Questions: Chapter 4 Practice: Introduction to Probability (i) The probabilities that stock A will rise in price is 0.40 and that stock B will rise in price is 0.60. Further, if stock B rises in price, the probability that stock A will also rise in price is 0.50. a. What is the probability that at least one of the stocks will rise in price? (Round your answer to 2 decimal places.) Probability b. Are events A and B mutually exclusive? - Yes because P(A B)=P(A). - Yes because P(A ∩ B)=0. - No because P(A B) ≠ P(A). - No because P(A ∩ B) ≠ 0. c. Are events A and B independent? - Yes because P(A B)=P(A). - Yes because P(A ∩ B)=0. - No because P(A B) ≠ P(A). - No because P(A ∩ B) ≠ 0.

Chapter 4 Practice: Introduction to Probability (i) The probabilities that stock A will rise in price is 0.40 and that stock B will rise in price is 0.60. Further, if stock B rises in price, the probability that stock A will also rise in price is 0.50. a. What is the probability that at least one of the stocks will rise in price? (Round your answer to 2 decimal places.) Probability b. Are events A and B mutually exclusive? - Yes because P(A B)=P(A). - Yes because P(A ∩ B)=0. - No because P(A B) ≠ P(A). - No because P(A ∩ B) ≠ 0. c. Are events A and B independent? - Yes because P(A B)=P(A). - Yes because P(A ∩ B)=0. - No because P(A B) ≠ P(A). - No because P(A ∩ B) ≠ 0.
Transcript text: Chapter 4 Practice: Introduction to Probability (i) 6 The probabilities that stock $A$ will rise in price is 0.40 and that stock $B$ will rise in price is 0.60 . Further, if stock $B$ rises in price, the probability that stock $A$ will also rise in price is 0.50 . 10 a. What is the probability that at least one of the stocks will rise in price? (Round your answer to 2 decimal places.) $\square$ Probability b. Are events $A$ and $B$ mutually exclusive? Yes because $P(A \mid B)=P(A)$. Yes because $P(A \cap B)=0$. No because $P(A \mid B) \neq P(A)$. No because $P(A \cap B) \neq 0$. c. Are events $A$ and $B$ independent? Yes because $P(A \mid B)=P(A)$. Yes because $P(A \cap B)=0$. No because $P(A \mid B) \neq P(A)$. No because $P(A \cap B) \neq 0$. Prev 6 of 10 Score answer >
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Solution

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Solution Steps

Solution Approach

a. To find the probability that at least one of the stocks will rise in price, use the formula for the probability of the union of two events: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Calculate \( P(A \cap B) \) using the conditional probability \( P(A \mid B) \).

b. Events are mutually exclusive if they cannot both occur at the same time, i.e., \( P(A \cap B) = 0 \). Check if this condition holds.

c. Events are independent if the occurrence of one does not affect the probability of the other, i.e., \( P(A \mid B) = P(A) \). Check if this condition holds.

Step 1: Calculate the Probability of Both Stocks Rising

To find the probability that both stocks \( A \) and \( B \) will rise, use the conditional probability formula: \[ P(A \cap B) = P(A \mid B) \times P(B) = 0.50 \times 0.60 = 0.30 \]

Step 2: Calculate the Probability of At Least One Stock Rising

The probability that at least one of the stocks will rise is given by the formula for the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.40 + 0.60 - 0.30 = 0.70 \] Thus, the probability that at least one stock will rise is \( \boxed{0.70} \).

Step 3: Determine if Events \( A \) and \( B \) are Mutually Exclusive

Events \( A \) and \( B \) are mutually exclusive if \( P(A \cap B) = 0 \). Since \( P(A \cap B) = 0.30 \), the events are not mutually exclusive. Therefore, the answer is \(\boxed{\text{No because } P(A \cap B) \neq 0}\).

Step 4: Determine if Events \( A \) and \( B \) are Independent

Events \( A \) and \( B \) are independent if \( P(A \mid B) = P(A) \). Given \( P(A \mid B) = 0.50 \) and \( P(A) = 0.40 \), these probabilities are not equal, indicating that the events are not independent. Therefore, the answer is \(\boxed{\text{No because } P(A \mid B) \neq P(A)}\).

Final Answer

  • Probability that at least one stock will rise: \(\boxed{0.70}\)
  • Are events \( A \) and \( B \) mutually exclusive? \(\boxed{\text{No because } P(A \cap B) \neq 0}\)
  • Are events \( A \) and \( B \) independent? \(\boxed{\text{No because } P(A \mid B) \neq P(A)}\)
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