If two dice are thrown together, then the probability that atleast one die will show a number greater than 4 is

Options

A. `2/3`
B. `1/3`
C. `4/9`
D. `5/9`

Answer

To solve the problem of finding the probability that at least one die will show a number greater than 4 when two dice are thrown together, we can follow these steps:

### Step 1: Determine the Total Outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ n(S) = 6 \times 6 = 36 \]

### Step 2: Identify Favorable Outcomes
Next, we need to find the number of outcomes where at least one die shows a number greater than 4. The numbers greater than 4 on a die are 5 and 6.

### Step 3: Calculate the Complement
Instead of counting the favorable outcomes directly, it is often easier to calculate the complement (the outcomes where neither die shows a number greater than 4). The numbers that are not greater than 4 are 1, 2, 3, and 4.

The number of outcomes where both dice show numbers less than or equal to 4 is:
\[ n(A) = 4 \times 4 = 16 \]
where \( A \) is the event that both dice show numbers less than or equal to 4.

### Step 4: Calculate the Number of Favorable Outcomes
The number of outcomes where at least one die shows a number greater than 4 is given by:
\[ n(B) = n(S) - n(A) = 36 - 16 = 20 \]
where \( B \) is the event that at least one die shows a number greater than 4.

### Step 5: Calculate the Probability
Now, we can calculate the probability of event \( B \):
\[ P(B) = \frac{n(B)}{n(S)} = \frac{20}{36} \]
This fraction can be simplified:
\[ P(B) = \frac{5}{9} \]

### Conclusion
Thus, the probability that at least one die will show a number greater than 4 is:
\[ \frac{5}{9} \]