Question

Let ${H_1},{H_2},....,{H_n}$be mutually exclusive and exhaustive events withP({H_i}) > 0,$$$$i = 1,2,.....n . Let $E$ be any other event with$0 < P(E) < 1$.
Statement 1: $P({H_i}|E) > P(E|{H_i}).P({H_i})$, for $i = 1,2,.....n$
Statement 2: $\sum\limits_{i = 1}^n {P({H_i}) = 1}$
A. Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
B. Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
C. Statement 1 is true, statement 2 is false.
D. Statement 1 is false, statement 2 is true.

Answer 1

If ${H_1},{H_2},....,{H_n}$be mutually exclusive and exhaustive events, then the sum of the probability is equal to one. Using this we will have statement I. we know the formula for conditional probability is $P\left( {\dfrac{A}{B}} \right) = \dfrac{{P(A \cap B)}}{{P(B)}}$. Using these we will say whether statement 1 and statement 2 is correct or not.

Complete step by step answer:
Given, ${H_1},{H_2},....,{H_n}$Is mutually exclusive and exhaustive events, sum of probability is equal to one, that is:
$\Rightarrow {H_1} \cup {H_2} \cup {H_3} \cup ........... \cup {H_n} = S$(Sample)
$\Rightarrow P({H_1}) + P({H_2}) + P({H_3}) + ......... + P({H_n}) = 1$
If we express in the summation form we have,
$\Rightarrow \sum\limits_{i = 1}^n {P({H_i}) = 1}$.
Hence statement 2 is true.
Let’s take the statement 1, that is
$P\left( {{H_i}|E} \right) > P\left( {E|{H_i}} \right) \times P({H_i})$
By the definition of conditional probability, we have $P\left( {A|B} \right) = \dfrac{{P(A \cap B)}}{{P(B)}}$, applying in above we get:
$\Rightarrow \dfrac{{P({H_i} \cap E)}}{{P(E)}} > \dfrac{{P(E \cap {H_i})}}{{P({H_i})}} \times P({H_i})$
Cancelling $P({H_i})$ we get,
$\Rightarrow \dfrac{{P({H_i} \cap E)}}{{P(E)}} > P(E \cap {H_i})$
Given, $0 < P(E) < 1$
It is obvious that $\dfrac{1}{{P(E)}}$is larger, that is greater than 1
Hence, $\dfrac{{P({H_i} \cap E)}}{{P(E)}} > P(E \cap {H_i})$ is true.
Hence statement 1 is correct.
Since statement 1 and statement 2 are not related. No statement is an explanation of the other.
Hence, the most convenient answer is option (B).

Note: Conditional probability refers to the chances that some outcome occurs given that another event has also occurred. Two events A and B are said to be mutually exclusive if the occurrence of A prohibits the occurrence of B (Vice versa). Two events A and B are said to be exhaustive if at least one of them will definitely occur.