Four dice are thrown simultaneously. If the occurrence of 2, 4 or 6 in single die is considered a success, find the probability of at least three successes.

Answer

To solve the problem of finding the probability of getting at least three successes when four dice are thrown, where a success is defined as rolling a 2, 4, or 6, we can follow these steps:

### Step 1: Define the Success Probability
When a single die is thrown, the successful outcomes are 2, 4, or 6. Therefore, the probability of success (P) is:
\[ P = \frac{\text{Number of successful outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]

### Step 2: Define the Failure Probability
The probability of failure (Q), which is rolling a 1, 3, or 5, is:
\[ Q = 1 - P = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 3: Identify the Number of Trials
We are throwing 4 dice, so the number of trials (n) is:
\[ n = 4 \]

### Step 4: Calculate the Probability of At Least 3 Successes
We need to find the probability of getting at least 3 successes (i.e., either 3 successes or 4 successes). This can be expressed as:
\[ P(X \geq 3) = P(X = 3) + P(X = 4) \]

### Step 5: Calculate \( P(X = 3) \)
Using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} P^k Q^{n-k} \]
For \( k = 3 \):
\[ P(X = 3) = \binom{4}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{4-3} = \binom{4}{3} \left(\frac{1}{2}\right)^4 \]
Calculating \( \binom{4}{3} = 4 \):
\[ P(X = 3) = 4 \cdot \left(\frac{1}{2}\right)^4 = 4 \cdot \frac{1}{16} = \frac{4}{16} = \frac{1}{4} \]

### Step 6: Calculate \( P(X = 4) \)
For \( k = 4 \):
\[ P(X = 4) = \binom{4}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{4-4} = \binom{4}{4} \left(\frac{1}{2}\right)^4 \]
Calculating \( \binom{4}{4} = 1 \):
\[ P(X = 4) = 1 \cdot \left(\frac{1}{2}\right)^4 = 1 \cdot \frac{1}{16} = \frac{1}{16} \]

### Step 7: Combine the Probabilities
Now, we can combine the probabilities of getting 3 and 4 successes:
\[ P(X \geq 3) = P(X = 3) + P(X = 4) = \frac{1}{4} + \frac{1}{16} \]
To add these fractions, we need a common denominator. The least common multiple of 4 and 16 is 16:
\[ P(X \geq 3) = \frac{4}{16} + \frac{1}{16} = \frac{5}{16} \]

### Final Answer
The probability of getting at least 3 successes when 4 dice are thrown is:
\[ \boxed{\frac{5}{16}} \]