What is the probability that 4 rolls of fair die will show three 4's?

the answer is 0.015, but I don't know how to get it.

To calculate the probability of obtaining a specific outcome in a series of independent events, such as rolling fair dice, you need to consider two factors: the number of favorable outcomes and the total number of possible outcomes.

In this case, we want to determine the probability of rolling three 4's in four rolls of a fair die.

Step 1: Determine the number of favorable outcomes:
To get three 4's in four rolls, we need to roll a 4 three times and any other number once.
The probability of getting a 4 on a single roll of a fair die is 1 out of 6 (as there are 6 possible outcomes - numbers 1 to 6 - and only one of them is a 4).
Therefore, the number of favorable outcomes is calculated as:

Number of favorable outcomes = (number of ways to get three 4's) * (number of ways to get any other number in the remaining roll)

To determine the number of ways to achieve these outcomes, we use binomial coefficients. The number of ways to choose three rolls out of four to be 4's is given by the binomial coefficient (4 choose 3), which is denoted as C(4, 3) or 4C3. This is calculated as:

4C3 = 4! / (3! * (4-3)!) = 4

Similarly, the number of ways to choose one roll to be any number other than 4 out of four rolls is 4C1, which is:

4C1 = 4! / (1! * (4-1)!) = 4

Therefore, the number of favorable outcomes is 4 * 4 = 16.

Step 2: Determine the total number of possible outcomes:
In each roll, there are 6 possible outcomes (numbers 1 to 6). Since we are rolling the die four times independently, the total number of possible outcomes is calculated as:

Total number of possible outcomes = (number of outcomes in each roll) ^ (number of rolls)

Total number of possible outcomes = 6^4 = 1296

Step 3: Calculate the probability:
Finally, we calculate the probability of getting three 4's in four rolls by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 16 / 1296
Probability ≈ 0.0123

The correct probability is approximately 0.0123, not 0.015.