Question

The probabilities that each of two independent experiments will have a successful outcome are \(\frac{8}{15}\) and \(\frac{2}{3}\), respectively. What is the probability that both experiments will have successful outcomes?

• A. \(\frac{4}{5}\)
• B. \(\frac{6}{5}\)
• C. \(\frac{2}{15}\)
• D. \(\frac{16}{45}\)
• E. \(\frac{64}{225}\)

Hint:

In probability problems, the word "and" is a key indicator for multiplication, especially when dealing with independent events. The word "or" typically indicates addition (with an adjustment for any overlap).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question deals with the probability of independent events. Two events are independent if the outcome of one does not affect the outcome of the other. To find the probability that both independent events occur, we multiply their individual probabilities.
Step 2: Key Formula or Approach:
If \(A\) and \(B\) are two independent events, the probability that both \(A\) and \(B\) occur is given by the formula:
\[ P(A \text{ and } B) = P(A) \times P(B) \] Step 3: Detailed Explanation:
Let \(P(S_1)\) be the probability of success for the first experiment and \(P(S_2)\) be the probability of success for the second experiment.
We are given:
\[ P(S_1) = \frac{8}{15} \] \[ P(S_2) = \frac{2}{3} \] We need to find the probability that both experiments are successful. Since the experiments are independent, we multiply their probabilities:
\[ P(S_1 \text{ and } S_2) = P(S_1) \times P(S_2) = \frac{8}{15} \times \frac{2}{3} \] Now, we multiply the numerators together and the denominators together:
\[ \frac{8 \times 2}{15 \times 3} = \frac{16}{45} \] Step 4: Final Answer:
The probability that both experiments will have successful outcomes is \(\frac{16}{45}\).