A die is thrown 3 times. Getting a multiple of 3 is considered a success. Find the probability of at least 2 successes.

Answer

To solve the problem of finding the probability of getting at least 2 successes when a die is thrown 3 times, where a success is defined as rolling a multiple of 3, we can follow these steps:

### Step 1: Identify the Probability of Success (P) and Failure (Q)

1. **Determine the successful outcomes**: The multiples of 3 on a die are 3 and 6. Therefore, there are 2 successful outcomes.
2. **Total outcomes when rolling a die**: There are 6 possible outcomes (1, 2, 3, 4, 5, 6).
3. **Calculate the probability of success (P)**:
\[ P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3} \]
4. **Calculate the probability of failure (Q)**:
\[ Q = 1 - P = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 2: Define the Number of Trials (n)

- The die is thrown 3 times, so the number of trials \( n = 3 \).

### Step 3: Use the Binomial Probability Formula

The probability of getting exactly \( r \) successes in \( n \) trials is given by the binomial probability formula:
\[ P(X = r) = \binom{n}{r} P^r Q^{n-r} \]
Where:
- \( \binom{n}{r} \) is the binomial coefficient (n choose r),
- \( P \) is the probability of success,
- \( Q \) is the probability of failure.

### Step 4: Calculate the Probability of At Least 2 Successes

We need to find the probability of getting at least 2 successes, which can be expressed as:
\[ P(X \geq 2) = P(X = 2) + P(X = 3) \]

#### Calculate \( P(X = 2) \):
\[ P(X = 2) = \binom{3}{2} P^2 Q^{3-2} = \binom{3}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^1 \]
Calculating the binomial coefficient:
\[ \binom{3}{2} = 3 \]
Now substitute:
\[ P(X = 2) = 3 \cdot \left(\frac{1}{3}\right)^2 \cdot \left(\frac{2}{3}\right) = 3 \cdot \frac{1}{9} \cdot \frac{2}{3} = 3 \cdot \frac{2}{27} = \frac{6}{27} \]

#### Calculate \( P(X = 3) \):
\[ P(X = 3) = \binom{3}{3} P^3 Q^{3-3} = \binom{3}{3} \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^0 \]
Calculating the binomial coefficient:
\[ \binom{3}{3} = 1 \]
Now substitute:
\[ P(X = 3) = 1 \cdot \left(\frac{1}{3}\right)^3 \cdot 1 = \frac{1}{27} \]

### Step 5: Combine the Probabilities

Now, we can combine the probabilities:
\[ P(X \geq 2) = P(X = 2) + P(X = 3) = \frac{6}{27} + \frac{1}{27} = \frac{7}{27} \]

### Final Answer
The probability of getting at least 2 successes when a die is thrown 3 times is:
\[ \frac{7}{27} \]