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Given that events A and B are independent with P(A)=0.06 and P(A and B)=0.054, determine the value of P(B), rounding to the nearest thousandth, if necessary. Answer:

Given that events A and B are independent with P(A)=0.06 P(A)=0.06 and P(A P(A and B)=0.054 B)=0.054 , determine the value of P(B) P(B) , rounding to the nearest thousandth, if necessary.\newlineAnswer:

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Q. Given that events A and B are independent with P(A)=0.06 P(A)=0.06 and P(A P(A and B)=0.054 B)=0.054 , determine the value of P(B) P(B) , rounding to the nearest thousandth, if necessary.\newlineAnswer:
  1. Independent Events Formula: Since events AA and BB are independent, the probability of AA and BB occurring together, P(A and B)P(A \text{ and } B), is equal to the product of their individual probabilities, P(A)×P(B)P(A) \times P(B).
  2. Calculate P(B)P(B): We are given P(A)=0.06P(A) = 0.06 and P(A and B)=0.054P(A \text{ and } B) = 0.054. We can use these values to find P(B)P(B) by rearranging the formula for independent events: P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B).
  3. Substitute Values: To find P(B)P(B), we divide P(A and B)P(A \text{ and } B) by P(A)P(A): P(B)=P(A and B)P(A)P(B) = \frac{P(A \text{ and } B)}{P(A)}.
  4. Perform Division: Substitute the given values into the equation: P(B)=0.0540.06P(B) = \frac{0.054}{0.06}.
  5. Final Probability: Perform the division to find P(B)P(B): P(B)=0.9P(B) = 0.9.
  6. Round to Nearest Thousandth: Since the problem asks for the probability rounded to the nearest thousandth, we do not need to round further because 0.90.9 is already at the thousandth place.

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