A pair of dice is thrown 7 times. If getting a total of 7 is considered a success, what is the probability of atmost 6 success?

Answer

To solve the problem of finding the probability of getting at most 6 successes when a pair of dice is thrown 7 times, we can follow these steps:

### Step 1: Determine the probability of success
When rolling a pair of dice, the total number of outcomes is 36 (since each die has 6 faces, and \(6 \times 6 = 36\)). To find the probability of getting a total of 7, we need to identify the favorable outcomes that yield this sum. The combinations that give a total of 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

Thus, there are 6 favorable outcomes.

**Probability of success (p)**:
\[ p = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

### Step 2: Determine the probability of failure
The probability of failure (not getting a total of 7) is:
\[ q = 1 - p = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 3: Use the binomial distribution
Since we are throwing the dice 7 times, we can model this situation using the binomial distribution. The number of successes (getting a total of 7) in 7 trials can be represented as \(X\), where \(X\) follows a binomial distribution with parameters \(n = 7\) and \(p = \frac{1}{6}\).

### Step 4: Calculate the probability of at most 6 successes
To find the probability of at most 6 successes, we can use the complement rule:
\[ P(X \leq 6) = 1 - P(X = 7) \]

### Step 5: Calculate \(P(X = 7)\)
Using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
For \(k = 7\):
\[ P(X = 7) = \binom{7}{7} \left(\frac{1}{6}\right)^7 \left(\frac{5}{6}\right)^{7-7} \]
\[ = 1 \cdot \left(\frac{1}{6}\right)^7 \cdot 1 = \left(\frac{1}{6}\right)^7 \]

### Step 6: Substitute back into the complement formula
Now, substituting back into the equation for \(P(X \leq 6)\):
\[ P(X \leq 6) = 1 - P(X = 7) = 1 - \left(\frac{1}{6}\right)^7 \]

### Final Answer
Thus, the probability of getting at most 6 successes when a pair of dice is thrown 7 times is:
\[ P(X \leq 6) = 1 - \frac{1}{6^7} \]