Question

Let S be the sample space of the random experiment of throwing simultaneously two unbiased dice and E_k={(a,b)∈S:ab=k}. If P_k=P(E_k), then the correct among the following is 

• A. p₁<p₁₀<p₄
• B. p₂<p₈<p₁₄
• C. p₄<p₈<p₁₇
• D. p₁<p₁₀<p₄

Answer 1

The Correct Option is A

The sample space $S$ for rolling two six-sided dice includes all ordered pairs $(x, y)$ where $x, y \in \{1, 2, 3, 4, 5, 6\}$.

Total outcomes in $S$: $6 \times 6 = 36$ 

Each outcome is equally likely. So, the probability of any one outcome (like $(1,1)$) is:

$P(1,1) = \frac{1}{36}$

Now, consider the event where the sum of dice is $i$, denoted as $p_i = P(x+y = i)$

Number of ways to get sum = $i$:

• $p_1 = 0$ (impossible)
• $p_2 = \frac{1}{36}$ (only (1,1))
• $p_3 = \frac{2}{36}$ ((1,2), (2,1))
• $p_4 = \frac{3}{36}$ ((1,3), (2,2), (3,1))
• $p_{10} = \frac{3}{36}$ ((4,6), (5,5), (6,4))

So the values are:

$p_1 = 0$, $p_{10} = \frac{3}{36}$, $p_4 = \frac{3}{36}$

Thus, the correct inequality is: $p_1 < p_{10} = p_4$

Correct option: (A): $p_1 < p_{10} < p_4$