A certain jar contains only b black marbles, w white marbles and r red marbles. If one marble is to...

Understanding the Question

We need to determine: Is the probability of drawing a red marble greater than the probability of drawing a white marble?

Since we're randomly selecting from all marbles in the jar, the probability of getting a red marble is simply \(\frac{\mathrm{r}}{\mathrm{b+w+r}}\), and the probability of getting a white marble is \(\frac{\mathrm{w}}{\mathrm{b+w+r}}\).

Notice that these probabilities have the same denominator. So comparing \(\mathrm{P(red)} > \mathrm{P(white)}\) is equivalent to comparing their numerators: Is \(\mathrm{r} > \mathrm{w}\)?

This is a yes/no question. To be sufficient, a statement must allow us to definitively answer either "yes, \(\mathrm{r} > \mathrm{w}\)" or "no, \(\mathrm{r} ≤ \mathrm{w}\)" — not just show that one is possible.

Analyzing Statement 1

Statement 1 tells us: \(\frac{\mathrm{r}}{\mathrm{b+w}} > \frac{\mathrm{w}}{\mathrm{b+r}}\)

This compares two ratios:

• Left side: The fraction of red marbles among all non-red marbles
• Right side: The fraction of white marbles among all non-white marbles

Let's explore what this inequality reveals about r versus w.

First, let's check if r could equal w:
If \(\mathrm{r} = \mathrm{w}\), then both fractions would have identical numerators. Since \(\mathrm{b+w}\) and \(\mathrm{b+r}\) would both equal b plus that common value, the fractions would be equal. But Statement 1 says the left side is strictly greater, so r cannot equal w.

Now let's test which direction the inequality goes:

Test Case 1 - When \(\mathrm{r} > \mathrm{w}\):

• Let r = 10, w = 5, b = 20
• Left side: \(\frac{10}{20+5} = \frac{10}{25} = 0.4\)
• Right side: \(\frac{5}{20+10} = \frac{5}{30} ≈ 0.167\)
• Check: Is 0.4 > 0.167? ✓ YES, the inequality holds!

Test Case 2 - When \(\mathrm{r} < \mathrm{w}\):

• Let r = 5, w = 10, b = 20
• Left side: \(\frac{5}{20+10} = \frac{5}{30} ≈ 0.167\)
• Right side: \(\frac{10}{20+5} = \frac{10}{25} = 0.4\)
• Check: Is 0.167 > 0.4? ✗ NO, the inequality fails!

The key insight: Statement 1's inequality can only be satisfied when \(\mathrm{r} > \mathrm{w}\). The cross-ratio comparison forces this specific relationship.

Since Statement 1 guarantees \(\mathrm{r} > \mathrm{w}\), we can definitively answer "YES" to our question.

[STOP - Statement 1 is Sufficient!]

Statement 1 is SUFFICIENT.

This eliminates answer choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(\mathrm{b} - \mathrm{w} > \mathrm{r}\)

This means the excess of black marbles over white marbles is greater than the number of red marbles. We can rearrange this as: \(\mathrm{b} > \mathrm{w} + \mathrm{r}\).

But notice — this constrains the relationship between black marbles and the sum of white and red, not the relationship between r and w directly.

Let's test whether both \(\mathrm{r} > \mathrm{w}\) and \(\mathrm{r} < \mathrm{w}\) are possible:

Scenario 1 - Where \(\mathrm{r} > \mathrm{w}\):

• Let b = 20, w = 5, r = 10
• Check Statement 2: Is 20 - 5 > 10? Is 15 > 10? ✓ YES
• And indeed \(\mathrm{r} > \mathrm{w}\) (10 > 5)

Scenario 2 - Where \(\mathrm{r} < \mathrm{w}\):

• Let b = 20, w = 10, r = 5
• Check Statement 2: Is 20 - 10 > 5? Is 10 > 5? ✓ YES
• But now \(\mathrm{r} < \mathrm{w}\) (5 < 10)

Since Statement 2 is satisfied in both scenarios — one where \(\mathrm{r} > \mathrm{w}\) and another where \(\mathrm{r} < \mathrm{w}\) — we cannot determine which probability is greater.

Statement 2 is NOT SUFFICIENT.

This eliminates answer choices B and D.

The Answer: A

Statement 1 alone is sufficient because the inequality \(\frac{\mathrm{r}}{\mathrm{b+w}} > \frac{\mathrm{w}}{\mathrm{b+r}}\) can only be true when \(\mathrm{r} > \mathrm{w}\), giving us a definitive "yes" to our question.

Statement 2 alone is not sufficient because it allows both \(\mathrm{r} > \mathrm{w}\) and \(\mathrm{r} < \mathrm{w}\) scenarios.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."