Question

Suppose that a sample space consists of ? equally likely outcomes. Select all of the statements that must be true.

a. Each outcome in the sample space has equal probability of occurring.

b. Any two events in the sample space have equal probablity of occurring.

c. The probability of any event occurring is the number of ways the event can occur divided by ?.

d. Probabilities can be assigned to outcomes in any manner as long as the sum of probabilities of all outcomes in the sample space is 1.

Answer 1

GIVEN:

Suppose that a sample space consists of ? equally likely outcomes.

a) Each outcome in the sample space has equal probability of occurring.

The statement (a) is TRUE.

The outcomes of a sample space are called equally likely if all of them have the same chance (equal probability) of occurring.

EXAMPLE:

• Throwing a Dice: There are six “6” possible outcomes when rolling a single dice. In this case all the six outcomes are assumed to be equally likely outcome. Each has a probability of occurrence 1/6.
• Playing a Card: There are 52 cards in a deck of ordinary playing cards. All the cards are of the same size and are therefore assumed equally likely to be chosen. Each card has a probability of occurence 1/52.
• Getting an even number on the toss of a die and getting an odd number on the toss of a die are equally likely events, since the probabilities of each event are equal.

b) Any two events in the sample space have equal probablity of occurring.

The statement (b) is FALSE.

​For a sample space consisting of n equally likely outcomes, each of the outcomes have equal probability of occurence and not every two outcomes have equal probability of occurence.

c) The probability of any event occurring is the number of ways the event can occur divided by ?.

The statement (c) is TRUE.

Suppose an event E can happen in r ways out of a total of n possible equally likely ways.

Then the probability of occurrence of the event (called its success) is denoted by,

   P[E] = r / n

EXAMPLE:

Getting a 5 if I roll a die.

A die has 6 equally likely outcomes ​.

There is only one 5 on a die. Thus n=6 and r=1

Thus the probability of getting a 5 is given by:

   P[E=5] = r / n

   = 1/6

Thus the probability of any event occurring is the number of ways the event can occur divided by ?.​

​d) Probabilities can be assigned to outcomes in any manner as long as the sum of probabilities of all outcomes in the sample space is 1.

The statement (d) is TRUE.

EXAMPLE:​

Throwing a Dice: There are six “6” possible outcomes when rolling a single die .

In this case all the six outcomes are assumed to be equally likely outcome.

Now we can assign probability 1/6 to each of these 6 outcomes ​ in any order or manner but the sum of probabilities of these outcomes in the sample space is 1.

  

Thus Options (a), (c) and (d) are true statements.