Question

Five balls, numbered 1, 2, 3, 4, and 5, are placed in an urn. Two balls are randomly selected from the five, and their numbers noted. Find the probability distribution for the following:
a) The largest of the two sampled numbers
b) The sum of the two sampled numbers

Answer 1

Concepts and reason

The concept of probability and probability distribution function is used to solve this problem.

Probability distribution is an expression that links each outcomes of a statistical experiment with its probability of occurrence. The probability of any simple event is its relative frequency of occurrence.

Probability is the state or an extent to which given event is likely to happen. In general, the probability is defined as a number between and where implies the impossibility of an event to occur and implies the certainty of an event to occur.

The function called probability distribution function if it will determine the probabilities at various states of the random variable.

Counting rules make it possible to identify and count the sample points of an experiment. Such rules are useful because they help us understand what may happen when certain kinds of experiments occur.

Fundamentals

Probability distribution is a function which determines the probabilities of a random variable for the different values that the random variable can take. It can be defined as:

Probability distribution is used to model this problem. Consider that there are x favourable cases to an event E out of a total of n cases. Then the probability of that event is written as:

Count the number of experimental outcomes when n objects are to be selected from a larger set of N objects. This number is typically expressed as, given by

where,

a)

Find the number of sample points in the sample space.

Choosing 2 balls out of 5 balls,

List the sample space.

Let the random variable Y represents largest of the two sampled numbers.

There are 10 sample points, all equally likely, so

Therefore, the probability distribution for the largest of the two sampled numbers is

b)

Find the number of sample points in the sample space.

Choosing 2 balls out of 5 balls,

List the sample space.

Let the random variable X represents the sum of the two sampled numbers.

Therefore, the probability distribution for the sum of the two sampled numbers is

Ans: Part a

The probability distribution for the largest of the two sampled numbers (Y) is

Part b

The probability distribution for the sum of the two sampled numbers is