If events E and F are independent and each has a probability p of occurring, which of the following represents...

1. Translate the problem requirements: We need to find the probability that event E occurs AND event F does NOT occur, where both events are independent with probability \(\mathrm{p}\) each
2. Identify the component probabilities: Determine the individual probabilities for E occurring and F not occurring
3. Apply independence rule: Use the multiplication rule for independent events to find the joint probability
4. Verify through answer choice elimination: Confirm our result makes logical sense by checking boundary cases

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're looking for in everyday language. We have two events: E and F. Each event has the same chance (probability \(\mathrm{p}\)) of happening. The key word here is "independent" - this means that whether E happens has absolutely no effect on whether F happens, and vice versa.

We want to find the probability of a very specific scenario: E happens AND F does NOT happen. Think of it like flipping two coins where each has a probability \(\mathrm{p}\) of landing heads. We want the probability that the first coin lands heads while the second coin lands tails.

Process Skill: TRANSLATE - Converting the problem statement into clear mathematical understanding

2. Identify the component probabilities

Now let's identify the individual pieces we need:

• The probability that E occurs = \(\mathrm{p}\) (given in the problem)
• The probability that F does NOT occur = \(1 - \mathrm{p}\) (since if F has probability \(\mathrm{p}\) of occurring, it has probability \(1-\mathrm{p}\) of not occurring)

This is a fundamental principle in probability: if something has probability \(\mathrm{p}\) of happening, then it has probability \((1-\mathrm{p})\) of not happening. These two probabilities must add up to 1, which represents certainty.

3. Apply independence rule

Here's where the independence becomes crucial. When two events are independent, the probability that both conditions happen is simply the product of their individual probabilities.

In plain English: To find the probability that "E occurs AND F does not occur," we multiply:

• Probability E occurs: \(\mathrm{p}\)
• Probability F does not occur: \((1-\mathrm{p})\)

Therefore: Probability = \(\mathrm{p} \times (1-\mathrm{p}) = \mathrm{p}(1-\mathrm{p}) = \mathrm{p} - \mathrm{p}^2\)

Let's verify this makes sense with a concrete example: If \(\mathrm{p} = 0.6\), then:

• Probability E occurs = \(0.6\)
• Probability F does not occur = \(1 - 0.6 = 0.4\)
• Combined probability = \(0.6 \times 0.4 = 0.24 = \mathrm{p} - \mathrm{p}^2 = 0.6 - 0.36 = 0.24\) ✓

4. Verify through answer choice elimination

Let's check our answer \(\mathrm{p} - \mathrm{p}^2\) against the boundary cases:

When \(\mathrm{p} = 0\) (impossible events):

• Our formula: \(0 - 0^2 = 0\) ✓ (Makes sense: if E never happens, then "E occurs but F doesn't" never happens)

When \(\mathrm{p} = 1\) (certain events):

• Our formula: \(1 - 1^2 = 0\) ✓ (Makes sense: if F always happens, then "F doesn't occur" never happens)

When \(\mathrm{p} = 0.5\):

• Our formula: \(0.5 - 0.25 = 0.25\) ✓ (Makes sense: equal chance for all four combinations of E and F occurring/not occurring)

Looking at the answer choices, our result \(\mathrm{p} - \mathrm{p}^2\) matches exactly with choice (E).

Final Answer

The probability that E will occur but F will not occur is \(\mathrm{p} - \mathrm{p}^2\).

Answer: E) \(\mathrm{p} - \mathrm{p}^2\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "independent events": Students often confuse independent events with mutually exclusive events. They might think that if E and F are independent, then E occurring means F cannot occur, leading them to believe the answer should simply be \(\mathrm{p}\). Remember: independent means one event doesn't affect the other's probability, while mutually exclusive means both events cannot happen simultaneously.

2. Misreading the compound condition: Students may focus only on "E will occur" and miss the critical "but F will not occur" part of the question. This leads them to think the answer is just \(\mathrm{p}\) (probability of E occurring) rather than the probability of the specific combination where E happens AND F doesn't happen.

3. Confusing "F will not occur" with "F will occur": When setting up the problem, students might incorrectly identify the probability of F not occurring as \(\mathrm{p}\) instead of \((1-\mathrm{p})\), essentially solving for "E occurs and F occurs" rather than "E occurs and F does not occur."

Errors while executing the approach

1. Incorrect application of complement rule: Students may calculate the probability that F does not occur as \(\mathrm{p}\) instead of \((1-\mathrm{p})\). This fundamental error in applying the complement rule (where P(not F) = 1 - P(F)) leads to calculating \(\mathrm{p} \times \mathrm{p} = \mathrm{p}^2\) instead of the correct \(\mathrm{p} \times (1-\mathrm{p})\).

2. Algebraic expansion errors: When multiplying \(\mathrm{p} \times (1-\mathrm{p})\), students might make simple algebraic mistakes such as getting \(\mathrm{p} - \mathrm{p}\) instead of \(\mathrm{p} - \mathrm{p}^2\), or incorrectly expanding to get \(\mathrm{p} + \mathrm{p}^2\) or other incorrect expressions due to sign errors or distribution mistakes.

3. Forgetting to multiply probabilities: Some students correctly identify that E occurs with probability \(\mathrm{p}\) and F doesn't occur with probability \((1-\mathrm{p})\), but then add these probabilities instead of multiplying them, getting \(\mathrm{p} + (1-\mathrm{p}) = 1\), which clearly doesn't make sense in context.

Errors while selecting the answer

1. Selecting \(\mathrm{p}^2\) thinking it represents the compound event: Students who correctly understand they need a compound probability but make errors in execution might see \(\mathrm{p}^2\) among the choices and select it, thinking that squaring \(\mathrm{p}\) somehow represents the combination of two events, even though their calculation should have yielded \(\mathrm{p} - \mathrm{p}^2\).

2. Choosing \(1 - \mathrm{p}^2\) due to complement confusion: Students might calculate the correct expression \(\mathrm{p} - \mathrm{p}^2\) but then incorrectly think they need the complement of this result, leading them to select \(1 - \mathrm{p}^2\) as their final answer.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for \(\mathrm{p}\)

Let's use \(\mathrm{p} = 0.6\) (60% probability for each event). This is a logical choice because:

• It's between 0 and 1 (valid probability range)
• It's not an extreme value (like 0, 1, or 0.5) that might hide computational errors
• It makes calculations manageable while being representative

Step 2: Calculate component probabilities

• Probability that E occurs = \(\mathrm{p} = 0.6\)
• Probability that F does NOT occur = \(1 - \mathrm{p} = 1 - 0.6 = 0.4\)

Step 3: Apply independence rule

Since E and F are independent events:

P(E occurs AND F does not occur) = P(E occurs) × P(F does not occur)

= \(0.6 \times 0.4 = 0.24\)

Step 4: Test answer choices with \(\mathrm{p} = 0.6\)

1. \(\mathrm{p}^2 = (0.6)^2 = 0.36\) ❌
2. \(1 - \mathrm{p}^2 = 1 - 0.36 = 0.64\) ❌
3. \(\mathrm{p}^2 - 1 = 0.36 - 1 = -0.64\) ❌ (impossible probability)
4. \(\mathrm{p}^2 - \mathrm{p} = 0.36 - 0.6 = -0.24\) ❌ (impossible probability)
5. \(\mathrm{p} - \mathrm{p}^2 = 0.6 - 0.36 = 0.24\) ✓

Step 5: Verify with a second value

Let's confirm with \(\mathrm{p} = 0.3\):

• P(E occurs AND F does not occur) = \(0.3 \times (1 - 0.3) = 0.3 \times 0.7 = 0.21\)
• Testing choice E: \(\mathrm{p} - \mathrm{p}^2 = 0.3 - (0.3)^2 = 0.3 - 0.09 = 0.21\) ✓

The answer is consistently E) \(\mathrm{p} - \mathrm{p}^2\)