In a committee of $25$ members,each member is proficient either in Mathematics or in Statistics or in both. If $19$ of them are proficient in Mathematics and $16$ of them are proficient in Statistics,then the probability that a person selected at random from the committee is proficient in both is

The number of $4$-digit numbers that are less than or equal to $2800$ and are either divisible by $3$ or by $11$ is equal to $............$.

The smallest set $A$ such that $A \cup \{1, 2\} = \{1, 2, 3, 5, 9\}$ is

In a study about a pandemic,data of $900$ persons was collected. It was found that:
$190$ persons had symptom of fever,
$220$ persons had symptom of cough,
$220$ persons had symptom of breathing problem,
$330$ persons had symptom of fever or cough or both,
$350$ persons had symptom of cough or breathing problem or both,
$340$ persons had symptom of fever or breathing problem or both,
$30$ persons had all three symptoms (fever,cough and breathing problem).
If a person is chosen randomly from these $900$ persons,then the probability that the person has at most one symptom is . . . . .

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails alone})$.

$20$ teachers of a school either teach mathematics or physics. $12$ of them teach mathematics while $4$ teach both the subjects. Then the number of teachers teaching physics is