Question

An experiment has three possible outcomes, I, J, and K. The probabilities of the outcomes are 0.25, 0.35, and 0.40, respectively. If the experiment is to be performed twice and the successive outcomes are independent, what is the probability that K will not be an outcome either time?

• A. 0.36
• B. 0.40
• C. 0.60
• D. 0.64
• E. 0.80

Hint:

For "and" probabilities with independent events, you multiply. For "or" probabilities with mutually exclusive events, you add. The phrase "not K" is a clue to either add the other probabilities or use the complement rule (\(1 - P(K)\)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a probability problem involving independent events. We need to find the probability of an event (not getting K) happening twice in a row.
Step 2: Key Formula or Approach:
1. First, find the probability of the event "K does not happen" in a single experiment. This is the complement of the event "K happens". \( P(\text{not } K) = 1 - P(K) \). 2. Since the two experiments are independent, the probability of both events happening is the product of their individual probabilities. \( P(A \text{ and } B) = P(A) \times P(B) \).
Step 3: Detailed Explanation:
We are given the probabilities of the three outcomes: \( P(I) = 0.25 \) \( P(J) = 0.35 \) \( P(K) = 0.40 \) (As a check, the sum of probabilities is \(0.25 + 0.35 + 0.40 = 1.00\)). The event we are interested in for a single trial is "K will not be an outcome". The probability of this event, \( P(\text{not } K) \), can be calculated in two ways: Method 1: Using the complement rule. \[ P(\text{not } K) = 1 - P(K) = 1 - 0.40 = 0.60 \] Method 2: Summing the other probabilities. The outcome is not K if it is either I or J. \[ P(\text{not } K) = P(I) + P(J) = 0.25 + 0.35 = 0.60 \] So, the probability that K does not occur in one experiment is 0.60. The question asks for the probability that K will not be an outcome \textit{either time} in two successive, independent experiments. This means we want the probability of (not K on the first trial) AND (not K on the second trial). Since the trials are independent, we multiply their probabilities: \[ P(\text{not K on both}) = P(\text{not K on 1st}) \times P(\text{not K on 2nd}) \] \[ P(\text{not K on both}) = 0.60 \times 0.60 = 0.36 \] Step 4: Final Answer:
The probability that K will not be an outcome either time is 0.36.