A and B throw a dice. The probability that A's throw is not greater than B's, is

Options

A. `(5)/(12)`
B. `(7)/(12)`
C. `(1)/(6)`
D. `(1)/(2)`

Explanation

To find the probability that A's throw is not greater than B's when both A and B throw a dice, we can follow these steps:

### Step 1: Understand the Problem
A and B each throw a six-sided dice. We need to determine the probability that A's result is less than or equal to B's result.

### Step 2: Determine the Total Outcomes
When A and B throw a dice, each has 6 possible outcomes (1 through 6). Therefore, the total number of outcomes when both throw the dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

### Step 3: Identify Favorable Outcomes
We need to count the number of outcomes where A's throw is not greater than B's throw. We can analyze this based on the value of B's throw:

- If B throws 1: A can throw 1 (1 option)
- If B throws 2: A can throw 1 or 2 (2 options)
- If B throws 3: A can throw 1, 2, or 3 (3 options)
- If B throws 4: A can throw 1, 2, 3, or 4 (4 options)
- If B throws 5: A can throw 1, 2, 3, 4, or 5 (5 options)
- If B throws 6: A can throw 1, 2, 3, 4, 5, or 6 (6 options)

Now, we can sum these options to find the total number of favorable outcomes:
\[ 1 + 2 + 3 + 4 + 5 + 6 = 21 \]

### Step 4: Calculate the Probability
The probability that A's throw is not greater than B's throw is given by the ratio of favorable outcomes to total outcomes:
\[ \text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{21}{36} \]

### Step 5: Simplify the Probability
Now, we simplify the fraction:
\[ \frac{21}{36} = \frac{7}{12} \]

### Final Answer
Thus, the probability that A's throw is not greater than B's throw is:
\[ \frac{7}{12} \]

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