In a single throw of three dice , find the probability of
not getting the same number on all the dice .

Answer

To solve the problem of finding the probability of not getting the same number on all three dice when thrown, we can follow these steps:

### Step 1: Determine the Total Possible Outcomes
When throwing three dice, each die has 6 faces. Therefore, the total number of possible outcomes when throwing three dice is calculated as follows:

\[ \text{Total outcomes} = 6 \times 6 \times 6 = 6^3 = 216 \]

### Step 2: Determine the Favorable Outcomes for Getting the Same Number
Next, we need to find the number of favorable outcomes where all three dice show the same number. The possible outcomes for this scenario are:

- (1, 1, 1)
- (2, 2, 2)
- (3, 3, 3)
- (4, 4, 4)
- (5, 5, 5)
- (6, 6, 6)

There are a total of 6 outcomes where all three dice show the same number.

### Step 3: Calculate the Probability of Getting the Same Number
The probability of getting the same number on all three dice is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(\text{same number}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{216} = \frac{1}{36} \]

### Step 4: Calculate the Probability of Not Getting the Same Number
To find the probability of not getting the same number on all three dice, we can use the complement rule:

\[ P(\text{not same number}) = 1 - P(\text{same number}) = 1 - \frac{1}{36} = \frac{36 - 1}{36} = \frac{35}{36} \]

### Final Answer
Thus, the probability of not getting the same number on all three dice is:

\[ \frac{35}{36} \]

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