Question

Discuss the following concepts and give examples from everyday life in which you might encounter each concept. Hint: For instance, consider the “experiment” of arriving for class. Some possible outcomes are not arriving (that is, missing class), arriving on time, and arriving late.

(a) Sample space
(b) Probability assignment to a sample space. In your discussion be sure to include answers to the following questions.

(1) Is there more than one valid way to assign probabilities to a sample space? Explain and give an example.
(2) How can probability be estimated by relative frequencies? How can probability be computed if events are equally likely?

WRITE IN YOUR OWN WRODS IN TEXT NOT IN IMAGE TEXT SINCE I CANT READ SOME OF THE HANDWRITING

Answer 1

a.sample space:-a list of all possible outcomes of an experiment is called the sample space.it is denoted by S.

Experiment :: Toss a coin 3 times

S = {HHH,HHT,HTH,THH,HTT,THT,TTH,TTT } There are = 8 possible outcomes. This is an example of a discrete or countable sample space.

(b) Probability assignment to a sample space: in a random experiment after listing the sample space we need to assign number,called probabilities.

1)is there more than one valid way to assign probabilities to a sample space? Explain and give an example.

yes there are more than one way.

example:

to illustrate different assignments of probabilities suppose:first writting the sample space:

ram goes to a ice cream shop to order a single dip ice cream.this shop has 4 different ice cream flavors.which flavor will ram order?

		flavor
	pobability	vanilla chocolate orange mango

now considering all fact as of probability we consider some possible probability assignment to my problem.

case 1.

suppose ram has brought a box in which 10 slips are labelled vanilla,10 are chocolate,10 are orange and 10 are mango.he makes his ice cream choice by choosing a slip at random.here each flavor would have a probability:10/40=1/4.

		flavor
	pobability	vanilla chocolate orange mango 1/4 1/4 1/4 1/4

case 2:

lets consider a different set of probability base on different assumption about Rams taste preferences.ram donot like plain flavors but like fruit flavors a lot.in this case i would assign vanilla and chocolate each a probability 0,and design the two other flavors probabilities that sum to1.so the probability assignment be:

		flavor
	pobability	vanilla chocolate orange mango 0 0 0.6 0.4

case 3:

lets consider a probability assignment from the shopkeeper's view.suppose the shopkeeper has no idea which ice cream ram will order.he has kept a record of how many ice cream of each type has been ordered -of 50 ice cream ordered 20 are vanilla,15 are chocolate,5 are orange,10 are mango.he believe that ram has similar tastes as previous customers,then it would be reasonable to apply the frequency viewpoint to assign probability:

		flavor
	pobability	vanilla chocolate orange mango 20/50 15/50 5/50 10/50

here each of the probability assignments used a different viewpoint:

1st one :classical viewpoint(each of the 4 slips has equal probability of being selected)

2 nd one:subjective viewpoint( based on ram's preference of flavor)

3 rd one:frequency viewpoint(based on observed flavor preferences of previous 50 customers)

b2. How can probability be estimated by relative frequencies?

relative frequency= (number of trials that are successful) / (total number of trials)
Relative frequency tells how often anything is happening after dividing by the total number of outcomes. It is more an experimental concept than a theoretical one. In general we use the relative frequency concept in case of big number of trials. This can only be done practically and not theoretically. Since the concept is experimental so it is quite possible to get distinct relative frequency each time the experiment is repeated

ex: In a class of 50 students, 25 students go to school by school bus, 15 by car and 10 by walk. Find the relative frequencies.

Solution: Total number of students = 50

Number of students used school bus = 25

Number of students used car = 15

Number of students go by walk = 10

The relative frequencies are:

School bus = 25/50 = 0.50

Car = 15/50 = 0.30

Walk = 10/50 = 0.20

2nd part:How can probability be computed if events are equally likely?

For any sample space with N equally likely outcomes, we assign the probability 1/N to each outcome.(for ex if needed you can consider the 1st case of my probability assignment example)