While playing Yahtzee, Amanda rolls five dice and gets 1, 2, 4, 5, 6. She decides to keep the 1, 2, and 4, and reroll the 5 and 6.

After rerolling the 5 and 6, what is the probability for a large straight (1-2-3-4-5) or three 4's?
Also, what is the probability for a small straight (1-2-3-4 plus anything else)or a pair of three's?

There are 36 possible outcomes of throwing two dice. For a 1-2-3-4-5 "large straight", you would have to throw 3,6 or 6,3. That has a probability of 2/36 = 1/18. For three fours, you need to throw 4,4, which has only a 1/36 probability. Do the others the same way.

To calculate the probabilities for these outcomes in Yahtzee, we need to understand the number of possible outcomes and the number of favorable outcomes.

1. Large Straight (1-2-3-4-5) or Three 4's:
To get a large straight, we need to have either a 1-2-3-4-5 combination or three 4's.
- Total number of outcomes when rerolling two dice: There are 6 possible outcomes for each die, so the total number of outcomes when rerolling two dice is 6 * 6 = 36.
- Favorable outcomes for large straight: There is only one way to get a large straight (1-2-3-4-5).
- Favorable outcomes for three 4's: There are 5 choose 2 ways to choose the positions for the two remaining dice (5C2 = 10) multiplied by the possibilities for each dice, which is 6 * 6. Therefore, the number of favorable outcomes for three 4's is 10 * (6 * 6) = 360.

To calculate the probability, we divide the number of favorable outcomes by the total number of outcomes:

Probability for large straight or three 4's = (Number of favorable outcomes) / (Total number of outcomes)
Probability = (1 + 360) / 36
Probability = 361/36
Probability ≈ 10.03%

2. Small Straight (1-2-3-4 plus anything else) or Pair of three's:
To get a small straight, we need to have a 1-2-3-4 combination with any number on the fifth die, or we need a pair of three's with the remaining three dice showing any numbers.
- Total number of outcomes when rerolling two dice: 36 (as calculated above).
- Favorable outcomes for small straight: For each of the two possible outcomes for the fifth die (5 or 6), there are four possible ways to choose the position for it, resulting in 2 * 4 = 8 favorable outcomes.
- Favorable outcomes for pair of three's: We need three dice to show 3's, so we need to find the number of ways to select three positions on the dice (5C3 = 10) multiplied by the possibilities for each dice, which is 6 * 6 * 6. Therefore, the number of favorable outcomes for a pair of three's is 10 * (6 * 6 * 6) = 2160.

Probability for small straight or pair of three's = (Number of favorable outcomes) / (Total number of outcomes)
Probability = (8 + 2160) / 36
Probability = 2168/36
Probability ≈ 60.22%

So, the probabilities for a large straight or three 4's is approximately 10.03%, and the probabilities for a small straight or a pair of three's is approximately 60.22%.