To solve this problem, we need to calculate the probability of obtaining either a small straight or a large straight after re-rolling the two sixes. Let's break down the solution into parts:

1. Understanding the Goal

The goal is to roll a small straight or a large straight by re-rolling the two sixes. Here's what we know about the desired outcomes:

• Small straight: A sequence of exactly four consecutive numbers, such as $\{1, 2, 3, 4\}$, $\{2, 3, 4, 5\}$, or $\{3, 4, 5, 6\}$.
• Large straight: A sequence of exactly five consecutive numbers, which can either be $\{1, 2, 3, 4, 5\}$ or $\{2, 3, 4, 5, 6\}$.

2. Initial Setup

The initial roll was $\{2, 3, 4, 6, 6\}$, and we are re-rolling the two sixes in the hope of achieving a straight.

When we re-roll the two sixes, each die has six possible outcomes (1 through 6). So, there are a total of $6 \times 6 = 36$ possible outcomes for these two dice.

3. Counting Favorable Outcomes

Now, we need to determine how many of these 36 possible outcomes will result in either a small straight or a large straight. Let's consider each type of straight:

Case A: Small Straight

To achieve a small straight, we need exactly four dice in sequence. Since the original set includes $\{2, 3, 4\}$, we need one of the new rolls to fit with this sequence to form a small straight. The following are the possible sequences and corresponding rolls for small straights:

1. Sequence $\{1, 2, 3, 4\}$: We need at least one of the new dice to roll a 1.
2. Sequence $\{2, 3, 4, 5\}$: We need at least one of the new dice to roll a 5.
3. Sequence $\{3, 4, 5, 6\}$: We need at least one of the new dice to roll a 5 or 6.

Let’s calculate the probability for each case.

1. Probability of getting at least one 1 or one 5 in two rolls:
   • Probability of NOT rolling a 1 or 5 on a single roll: $\frac{4}{6} = \frac{2}{3}$
   • Probability of not rolling a