1.4 Equally Likely Outcomes

          The outcomes of a sample space are called equally likely if all of them have the same chance of occurrence. It is very difficult to decide whether or not the outcomes are equally likely. But in this tutorial we shall assume in most of the experiments that the outcomes are equally likely. We shall apply the assumption of equally likely in the following cases：

(1) Throw of a coin or coins: When a coin is tossed, it has two possible outcomes called

head and tail. We shall always assume that head and tail are equally likely if not otherwise mentioned. For more than one coin, it will be assumed that on all the coins, head and tail are equally likely

(2) Throw of a die or dice:       Throw of a single die can be produced six possible outcomes. All the six outcomes are assumed equally likely. For any number of dice, the six faces are assumed equally likely.

(3) Playing Cards:       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.

If an experiment has n simple outcomes, this methodwould assign a probability of 1/n to each outcome. Inother words, each outcome is assumed to have an equalprobability of occurrence.This method is also called the axiomatic approach.

Example 1: Roll of a DieS = {1, 2, · · · , 6}Probabilities: Each simple event has a 1/6 chance of occurring.

Example 2: Two Rolls of a DieS = {(1, 1), (1, 2), · · · , (6, 6)}Assumption: The two rolls are “independent.”Probabilities: Each simple event has a (1/6) · (1/6) =1/36 chance of occurring.

Theorem1.1In classical probability counting is used for calculatingprobabilities. For the probability of an event A we need to know the number of outcomes in A, k, andif the sample space consists of a finite number of equally likely outcomes, also the total number of outcomes, n.

Number of times when A occurs( ) lim lim

Number of total trialsA

N N

nP A

n

Number of elements in A ( )

Number of elements in S

kP A

n

E1:   A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color?

    

                  

   

Number of elements in A ( )

Number of elements in S

kP A

n

Number of ways to land on yellow 1P(yellow) =

Total number of colors 4

Number of ways to land on blue 1P(blue) =

Total number of colors 4

Number of ways to land on green 1P(green) =

Total number of colors 4

Number of ways to land on red 1P(red) =

Total numberof colors 4

E2:  

A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number?

Roll of a Die

P(even) = 3/6

P(low) = 3/6

P(even and low) = P({2}) = 1/6

P(even or low) = 3/6 + 3/6 − 1/6 = 5/6

P({1} or {6}) = 1/6 + 1/6 − 0 = 2/6

E3:   A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble?

                   

           

number of ways to choose yellow 3P(yellow) =

Total number of marbles 22

number of ways to choose blue 8 4P(blue) =

Total number of marbles 22 11

number of ways to choose green 5P(green) =

Total number of marbles 22

number of ways to choose red 6 3P(red) =

Total number of marbles 22 11