Question

2 dice are thrown in the air. Find the probability of getting a sum less than 11.

• A. 7/12
• B. 11/12
• C. 5/12
• D. 1/3

Hint:

Always use the complement rule \( 1 - P(\text{Not } A) \) when asked for "less than" or "greater than" probabilities that contain many terms.
The total number of outcomes is 36.
The only sums that are NOT less than 11 are 11 (2 ways) and 12 (1 way).
This gives 3 outcomes out of 36.
Subtract from 1: \( 1 - \frac{3}{36} = 1 - \frac{1}{12} = \frac{11}{12} \).
This takes very little time to solve!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question relates to the study of probability when rolling two fair, independent, six-sided dice.
The sum of the numbers showing on top of both dice can range from a minimum of 2 up to a maximum of 12.
We need to find the probability of getting a sum that is strictly less than 11.
Since counting all outcomes for sums less than 11 is tedious, it is much easier to use the complementary probability rule.

Step 2: Key Formula or Approach:

• Total number of outcomes when two dice are rolled is \( n(S) = 6 \times 6 = 36 \).
  
• The complementary probability formula is: \( P(A) = 1 - P(A') \), where $A'$ is the complement of event $A$.
  
• Here, event $A$ is "sum is less than 11" (i.e., \( \text{Sum} < 11 \)).
  
• The complement event $A'$ is "sum is 11 or greater" (i.e., \( \text{Sum} \ge 11 \)).
  
• The possible sums \( \ge 11 \) are $11$ and $12$.
  

Step 3: Detailed Explanation:

• First, let us identify the total number of outcomes in the sample space $S$ of rolling two six-sided dice:
  \[ n(S) = 6 \times 6 = 36 \]
  
• Now, let us define the complementary event $A'$, where the sum of the two dice is greater than or equal to 11.
  
• Let us list the specific outcomes that yield a sum of $11$:
  
• $(5, 6)$ and $(6, 5)$
  
• Let us list the specific outcomes that yield a sum of $12$:
  
• $(6, 6)$
  
• Thus, the total favorable outcomes for the complement event $A'$ are:
  \[ \text{Outcomes}(A') = \{(5,6), (6,5), (6,6)\} \]
  
• The number of favorable outcomes for $A'$ is:
  \[ n(A') = 3 \]
  
• Calculate the probability of the complementary event $P(A')$:
  \[ P(A') = \frac{n(A')}{n(S)} = \frac{3}{36} = \frac{1}{12} \]
  
• Now, calculate the probability of the target event $A$ (sum less than 11) using the complement rule:
  \[ P(A) = 1 - P(A') \]
  \[ P(A) = 1 - \frac{1}{12} \]
  \[ P(A) = \frac{12 - 1}{12} = \frac{11}{12} \]
  
• Thus, the probability of getting a sum less than 11 is $\frac{11}{12}$.
  

Step 4: Final Answer:
The probability of getting a sum less than 11 is $11/12$, which corresponds to Option (B).