Let S = {E1, E2, ……………., E8} be a sample space of a random experiment such that \(P(E_n) = \frac{n}{36}\) for every n = 1, 2, ………, 8. Then the number of elements in the set\(\{ A \subseteq S : P(A) \geq \frac{4}{5} \}\) is_________.
Correct Answer: 19
Solution and Explanation
Here \(P(E_n) = \frac{n}{36}\) for n = 1, 2, 3, ………, 8
Here,
\(P(A) = \frac{\text{Any possible sum of }(1,2,3,\ldots,8) \, (= a \, \text{say})}{36}\)
\(∵ \frac{a}{36}≥\frac{4}{5}\)
\(∴ a ≥ 29\)
If one of the number from {1, 2, …., 8} is left then total \(a ≥ 29\) by 3 ways.
Similarly by leaving terms more 2 or 3 we get 16 more combinations.
∴ Total number of different set A possible is \(16+3=19\)
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".