Question

Two different dice are rolled together. The probability that both the obtained numbers are less than 4, is

• A. \(\frac{2}{9}\)
• B. \(\frac{7}{36}\)
• C. \(\frac{1}{4}\)
• D. \(\frac{2}{3}\)

Hint:

Since rolling two separate dice are independent events, you can find the individual probabilities and multiply them:
- Probability of getting a number less than 4 on the first die is \(\frac{3}{6} = \frac{1}{2}\).
- Probability of getting a number less than 4 on the second die is \(\frac{3}{6} = \frac{1}{2}\).
- Since they are independent, the total probability is:
\[ P = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \] This product rule is extremely fast and prevents you from having to list out all the coordinate pairs!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The topic of this question is Probability.
When two fair, six-sided dice are rolled simultaneously, each die can show any integer from 1 to 6.
We need to determine the probability that both of the obtained numbers are strictly less than 4.

Step 2: Key Formula or Approach:
The probability of an event \(E\) is given by the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] We will find the total number of possible outcomes when two dice are rolled, list and count the outcomes that are favorable to our given condition, and then calculate the ratio.

Step 3: Detailed Explanation:

• Determine the total number of possible outcomes:
Since each die has 6 faces numbered 1 to 6, rolling two dice gives:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

• Identify the numbers on a single die that are strictly less than 4:
The numbers on a die that are less than 4 are:
\[ \{1, 2, 3\} \]

• List all possible pairs \((x, y)\) where both \(x\) (the number on the first die) and \(y\) (the number on the second die) belong to the set \(\{1, 2, 3\}\):
- \((1, 1), (1, 2), (1, 3)\)
- \((2, 1), (2, 2), (2, 3)\)
- \((3, 1), (3, 2), (3, 3)\)

• Count the number of favorable outcomes:
There are \(3 \times 3 = 9\) favorable outcomes.

• Calculate the probability of the event:
\[ P(\text{both numbers } \lt 4) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \] \[ P = \frac{9}{36} \]

• Simplify the fraction by dividing the numerator and denominator by 9:
\[ P = \frac{1}{4} \]

Step 4: Final Answer:
The probability that both obtained numbers are less than 4 is \(\frac{1}{4}\).
Therefore, the correct option is (C).