In a research study, each of 15 participants has been given exactly 1 of 15 printing devices. All 15 devices...

OWNING THE DATASET

Let's start by understanding what we're working with in this probability table. The dataset shows participants in an experiment, with each participant having:

• p: Their probability of success in a single attempt
• n: Number of attempts they were given
• Other data about their overall success probability

The key insight is that we need to quickly determine when a participant's overall probability of success is below 80%. Instead of calculating every value, we can recognize important probability thresholds:

• For p=0.1: Need at least 16 attempts to reach 80% success
• For p=0.2: Need at least 8 attempts to reach 80% success
• For p=0.5: Need at least 4 attempts to reach 80% success

This pattern recognition will help us evaluate participants much faster than calculating each probability.

QUESTION UNDERSTANDING

The question asks us to determine which participants contradict the hypothesis that "each participant had at least an 80% probability of overall success."

For each participant, we need to check if their probability of success across all attempts is at least 80%. The formula for this is:

\(\mathrm{P(success\ in\ n\ attempts)} = 1 - (1-\mathrm{p})^\mathrm{n}\)

Rather than calculating this formula each time, we'll use our optimized approach to quickly determine if each participant meets the threshold.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "Participant 6, with p=0.1 and n=8, contradicts the hypothesis."
What we're looking for:

• Does Participant 6 have less than an 80% probability of success?
• With p=0.1 and n=8, is the overall success probability < 80%?

In other words: Does a 10% chance per attempt over 8 attempts give less than 80% overall chance of success?

Let's apply our pattern recognition approach first:

• We know that with p=0.1, a participant needs at least 16 attempts to reach 80% success probability
• Participant 6 only had 8 attempts, which is less than the required 16 attempts
• Therefore, their success probability must be less than 80%

We can verify this with a quick calculation if needed:

\(\mathrm{P(success)} = 1 - (1-0.1)^8 = 1 - 0.9^8 = 1 - 0.43 = 0.57\) or 57%

Since 57% is less than 80%, Participant 6 does contradict the hypothesis.

Statement 1 CONTRADICTS THE HYPOTHESIS.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Participant 7, with p=0.5 and n=2, contradicts the hypothesis."
What we're looking for:

• Does Participant 7 have less than an 80% probability of success?
• With p=0.5 and n=2, is the overall success probability < 80%?

In other words: Does a 50% chance per attempt over 2 attempts give less than 80% overall chance of success?

Let's apply our standard probability patterns approach:

• For p=0.5, we know that 4 attempts are needed to reach 80% success
• Participant 7 only had 2 attempts, which is less than required
• This is actually a common probability pattern: with p=0.5 and n=2, the success probability is exactly 75%

The calculation is straightforward:

\(\mathrm{P(success)} = 1 - (1-0.5)^2 = 1 - 0.5^2 = 1 - 0.25 = 0.75\) or 75%

Since 75% is less than 80%, Participant 7 does contradict the hypothesis.

Statement 2 CONTRADICTS THE HYPOTHESIS.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "Participant 9, with p=0.26 and n=3, contradicts the hypothesis."
What we're looking for:

• Does Participant 9 have less than an 80% probability of success?
• With p=0.26 and n=3, is the overall success probability < 80%?

In other words: Does a 26% chance per attempt over 3 attempts give less than 80% overall chance of success?

This isn't a standard case, so let's use our approximation strategy:

• p=0.26 is between 0.2 and 0.5
• We know p=0.2 needs 8 attempts and p=0.5 needs 4 attempts to reach 80%
• Since 0.26 is closer to 0.2 than to 0.5, the required attempts should be closer to 8 than to 4
• With only 3 attempts, it's very likely below 80%

Let's do a quick approximation:

\(\mathrm{P(success)} = 1 - (1-0.26)^3 = 1 - 0.74^3\)

\(0.74^3 \approx 0.4\), so \(\mathrm{P(success)} \approx 1 - 0.4 = 0.6\) or 60%

Since 60% is less than 80%, Participant 9 does contradict the hypothesis.

Statement 3 CONTRADICTS THE HYPOTHESIS.

FINAL ANSWER COMPILATION

Reviewing our analysis:

• Statement 1: CONTRADICTS THE HYPOTHESIS - Participant 6 contradicts the hypothesis
• Statement 2: CONTRADICTS THE HYPOTHESIS - Participant 7 contradicts the hypothesis
• Statement 3: CONTRADICTS THE HYPOTHESIS - Participant 9 contradicts the hypothesis

The correct answer is: All three statements CONTRADICT THE HYPOTHESIS.

LEARNING SUMMARY

Skills We Used

• Pattern Recognition: We recognized standard probability thresholds instead of calculating every value
• Benchmarking: We compared participants' attempts to the required attempts for their success rate
• Strategic Approximation: For non-standard values, we used quick approximations rather than lengthy calculations

Strategic Insights

1. The "Attempt Gap" Method: Compare actual attempts to required attempts for 80% success - if actual < required, the participant contradicts the hypothesis
2. Know Your Benchmarks: Memorizing common probability thresholds saves significant time
3. Approximation Works: When dealing with non-standard probabilities, comparing to the nearest benchmark gives you a good starting point

Common Mistakes We Avoided

• Calculating Every Probability: We didn't need to fully calculate every participant's success probability
• Working Inefficiently: We used pattern recognition before calculation
• Missing Context: We considered the relationship between p and n, not just individual values

When solving probability problems on the GMAT, remember that you often don't need exact calculations. Identifying whether a value falls above or below a threshold is frequently enough, and pattern recognition can help you make these determinations much faster than detailed calculations.