MP Board · Class 10 · Mathematics · ProbabilityTwo dice are thrown simultaneously. What is the probability that the sum of the two numbers appearing on the top of the dice is (i) 8? (ii) 13? (iii) less than or equal to 12? Show all calculations clearly.

Step-by-Step Solution:

Step 1: Determine the total number of possible outcomes (Sample Space).\nWhen two dice are thrown simultaneously, each die has 6 faces numbered 1 to 6. $$\text{Total number of outcomes} = 6 \times 6 = 36$$\nLet the sample space be denoted by $S$, so $n(S) = 36$.\nThe sample space consists of ordered pairs $(x, y)$ where $x$ is the outcome on the first die and $y$ is the outcome on the second die.

Step 2: Find the probability that the sum of the two numbers is 8.

• Let $A$ be the event that the sum of the numbers is 8.
• Let us list all the favorable outcomes where the sum equals 8: $$A = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}$$
• Count the number of favorable outcomes: $n(A) = 5$. \nUsing the probability formula: $$P(A) = \frac{n(A)}{n(S)} = \frac{5}{36}$$

Step 3: Find the probability that the sum of the two numbers is 13.

• Let $B$ be the event that the sum of the numbers is 13.
• The maximum possible sum on two dice is $6 + 6 = 12$.
• Therefore, it is impossible to get a sum of 13.
• The set of favorable outcomes is an empty set: $B = \emptyset$.
• Count of favorable outcomes: $n(B) = 0$. \nUsing the probability formula: $$P(B) = \frac{n(B)}{n(S)} = \frac{0}{36} = 0$$

Step 4: Find the probability that the sum of the two numbers is less than or equal to 12.

• Let $C$ be the event that the sum of the numbers is less than or equal to 12.
• Since the minimum sum on two dice is $1 + 1 = 2$ and the maximum sum is $6 + 6 = 12$, the sum of any two numbers rolled on a pair of dice will always be between 2 and 12 inclusive.
• Therefore, every single outcome in the sample space satisfies this condition.
• The number of favorable outcomes: $n(C) = 36$. \nUsing the probability formula: $$P(C) = \frac{n(C)}{n(S)} = \frac{36}{36} = 1$$

Final Answer:

(i) Probability that the sum is 8 = $\frac{5}{36}$ (ii) Probability that the sum is 13 = $0$ (iii) Probability that the sum is less than or equal to 12 = $1$