Answer:
The required probability is
.
Step-by-step explanation:
Set given is:
S = {1, 2, 3, 5, 15, 21, 29, 38, 500}
Total number of elements in set,
= 9
Let A be the event that the number is less than 29 ({1, 2, 3, 5, 15, 21}).
Number of items in the event A,
= 6
Probability of event A,
![P(A) = \dfrac{n(A)}{n(S)}}=\dfrac{6}{9} \Rightarrow \dfrac{2}{3}](https://tex.z-dn.net/?f=P%28A%29%20%3D%20%5Cdfrac%7Bn%28A%29%7D%7Bn%28S%29%7D%7D%3D%5Cdfrac%7B6%7D%7B9%7D%20%5CRightarrow%20%5Cdfrac%7B2%7D%7B3%7D)
Formula for probability of any event E:
![P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}](https://tex.z-dn.net/?f=P%28E%29%20%3D%20%5Cdfrac%7B%5Ctext%7BNumber%20of%20favorable%20cases%7D%7D%7B%5Ctext%20%7BTotal%20number%20of%20cases%7D%7D)
Let B be the event that the number is odd (either of {1,3,5,15,21,29}).
Number of items in the event B,
= 6
Probability of event B,
![P(B) = \dfrac{n(B)}{n(S)}}=\dfrac{6}{9} \Rightarrow \dfrac{2}{3}](https://tex.z-dn.net/?f=P%28B%29%20%3D%20%5Cdfrac%7Bn%28B%29%7D%7Bn%28S%29%7D%7D%3D%5Cdfrac%7B6%7D%7B9%7D%20%5CRightarrow%20%5Cdfrac%7B2%7D%7B3%7D)
The event A and B have a few elements in common, i.e. numbers less than 29 which are odd as well.
The common elements are represented as:
![A \cap B = \{1, 3, 5, 15, 21\}](https://tex.z-dn.net/?f=A%20%5Ccap%20B%20%3D%20%5C%7B1%2C%203%2C%205%2C%2015%2C%2021%5C%7D)
![n(A\cap B) = 5](https://tex.z-dn.net/?f=n%28A%5Ccap%20B%29%20%3D%205)
![P(A \cap B ) = \dfrac{n(A \cap B)}{n(s)}\\\Rightarrow P(A \cap B) = \dfrac{5}{9}](https://tex.z-dn.net/?f=P%28A%20%5Ccap%20B%20%29%20%3D%20%5Cdfrac%7Bn%28A%20%5Ccap%20B%29%7D%7Bn%28s%29%7D%5C%5C%5CRightarrow%20P%28A%20%5Ccap%20B%29%20%3D%20%5Cdfrac%7B5%7D%7B9%7D)
To find probability of selecting a number which is either less than 29 (event A) or odd (event B),
We have to find
which is represented as
and the formula is:
![P(A \cup B) = P(A) + P(B) - P(A \cap B)\\\Rightarrow \dfrac{2}{3} + \dfrac{2}{3} - \dfrac{5}{9}\\\Rightarrow \dfrac{12-5}{9}\\\Rightarrow \dfrac{7}{9}](https://tex.z-dn.net/?f=P%28A%20%5Ccup%20B%29%20%3D%20P%28A%29%20%2B%20P%28B%29%20-%20P%28A%20%5Ccap%20B%29%5C%5C%5CRightarrow%20%5Cdfrac%7B2%7D%7B3%7D%20%2B%20%5Cdfrac%7B2%7D%7B3%7D%20-%20%5Cdfrac%7B5%7D%7B9%7D%5C%5C%5CRightarrow%20%5Cdfrac%7B12-5%7D%7B9%7D%5C%5C%5CRightarrow%20%5Cdfrac%7B7%7D%7B9%7D)
The required probability is
.