Question

Let $S$ be the sample space of the random experiment of throwing simultaneously two unbiased dice with six faces numbered $1$ to $6$ and Given,$E_k=\left\{\left(a,b\right)\in S:ab=k\right\}$ for $k\ge 1$. If $P_k=P\left(E_k\right)$ for $k\ge 1$, Then the correct among the following.

• A. $P_1<P_{30}<P_4<P_6$
• B. $P_{36}<P_6<P_2<P_4$
• C. $P_1<P_{11}<P_4<P_6$
• D. $P_{36}<P_{11}<P_6<P_4$

Answer 1

The correct option is D

$P_{36}<P_{11}<P_6<P_4$

Explanation for the correct answer :

Given sample space of random using conditional probability which states an event or outcome occurring based on the existence of a preceding event or outcome is known as conditional probability.
It's calculated by multiplying the likelihood of the previous occurrence by the probability of the next, or conditional, event.

The total number of elements in the sample space is $36$.

$$\begin{gathered} S=(1,1),(1,2),(1,3),(1,4),(1,5),(1,6) \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6) \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6) \\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6) \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6) \end{gathered}$$

From the above sample space,

Given, $E_k=\left\{\left(a,b\right)\in S:ab=k\right\}$

for, $P\left(E_1\right)$,$ab\ge 1,\left(a,b\right)=\left\{1,1\right\},\left\{1,2\right\},\dots \dots \dots \dots \left\{6,6\right\}$ all $36$ outcomes are possible

for, $P\left(E_2\right)$,$ab\ge 2,\left(a,b\right)=\left\{1,2\right\},\left\{1,3\right\},\dots \dots \dots \dots \left\{6,6\right\}$ all $35$ outcomes are possible

Hence, the probabilities will be

$$\begin{gathered} P\left(E_1\right)=1 \\ P\left(E_2\right)=\frac{35}{36} \\ P\left(E_3\right)=\frac{34}{36} \\ P\left(E_4\right)=\frac{33}{36} \end{gathered}$$

and so on, and it shows that probability will decrease as the value of $k$ will increase so

$P\left(E_1\right)>P\left(E_2\right)>P\left(E_3\right)>\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots >P\left(E_{36}\right)$

Thus, we see .$P_{36}<P_{11}<P_6<P_4$ is true.

Hence, the correct answer is option (D).