Given that a sample space is discrete if it contains either a finite number of points or a countably infinite nunber of points and a sample space is continuous if it contains uncountable number of points
1) if two distinct points on a circle radius 'r' are chosen at random and find the distance between them, write down the sample space
2) if a coin is tossed till a tail appears,find the sample space
1) The sample space for the distance between two distinct points on a circle with radius 'r' can be represented as a continuous interval. Let's assume that the circle is centered at the origin of a coordinate system. The sample space can be written as:
Sample space = [0, 2πr]
The interval [0, 2πr] represents all possible distances between the two points on the circumference of the circle.
2) The sample space for tossing a coin until a tail appears can be represented as a discrete set of positive integers, as the number of tosses required is finite but can be infinite (although with extremely low probability). The sample space can be written as:
Sample space = {1, 2, 3, 4, ...}
Each element in the set represents the number of coin tosses required until a tail appears.
1) To find the sample space for the distance between two distinct points on a circle, you need to consider all possible distances that can be obtained.
Let's assume that the circle has a radius 'r'. The distance between any two points on a circle can vary between 0 and 2*r (inclusive). This is because the minimum distance between two points on a circle occurs when the two points are right next to each other (0 distance), and the maximum distance occurs when the two points are directly opposite each other (2*r distance).
So, the sample space for this scenario can be represented as S = {0, 1, 2, 3, ..., 2r}, where each element in the set represents a possible distance between the two chosen points on the circle.
2) To find the sample space for tossing a coin until a tail appears, you need to consider all possible outcomes.
When tossing a fair coin, you have two possible outcomes on each toss: either a "head" or a "tail". The sample space for this scenario can be represented in terms of the number of tosses required to get a tail.
Let's denote "H" as a head and "T" as a tail. The sample space for this scenario can be represented as S = {T, HT, HHT, HHHT, ..., under bar notation (before the first tail appears)}.
The sample space is countably infinite since it contains an infinite number of elements, but each element can be represented using a finite sequence of heads and tails.