 # SAMPLE SPACE,EVENT SPACE ,INDEPENDENT AND DEPENDENT EVENTS.

Definition:

Probability: Is a measure of the likelihood  that an event will occur in any one trial. Probability can be applied in several areas like insurance, industrial quality control and so on.

It could also be defined as the ratio of required outcome to the total outcome.

Probability = No of required outcome

Total outcome

Sample Space: This is the set of all possible outcomes of any random experiment, and it’s denoted by S. The number of outcomes is denoted by n(S).

Event Space: This is the collection of outcomes of a random experiment. The number is denoted by n(E).

Outcome: This is result of an experiment in probability.

The probability that an event is certain to happen is 1, while the probability that an event is certain  not to happen is zero (0).

Range of inequality: 0 <pr (E)<1

Hence; Prob (an event will occur) + prob (an event will not occur) = 1.

EXAMPLE:

1.      From a box containing 2 red,6 white and 5 black balls, a ball is randomly selected . What is the Probability that the selected ball is (i) black (ii) white (iii) not black?

Solution:

Sample space = 2+ 6 + 5 = 13

n(red) = 2,       n(white) = 6,   n(black) = 5

(i)                 Prob. (black) = 5/13                       (ii)  Prob. (white) = 6/13

(iii)            Prob. (not black) = 1 – prob. (black)

(iv)                                         = 1 – 5/13 = 8/13

2.      What is the probability that an integer selected from the set of integers {20, 21, …, 30} is a prime number?

Evaluation: 1. A box contains five 10 ohm resistors and twelve 30 ohm resistor. The resistors are all unmarked and of the same physical size. If one resistor is picked out at random, determine the probability of its resistance being 10ohms.

EQUIPROBABLE SAMPLE SPACE:

Coin: A coin has two faces called the head (H) and the tail (T). The outcome of the experiment involving a coin depends on the numbers of trials.

In a single throw of a fair coin: {H, t} = 2

Two coins thrown once or a coin thrown twice: {HH, HT, TH, TT} = 4

Three coins of thrown thrice: {HHH, HHT, HTT, HTH, THH, THT, TTH, TTT} = 8.

Example: In a single throw of two fair coins. Find the probability that :  (i) two tails appear (ii) one head one tail appear (iii) two heads appear (iv) one tail one head in that order.

Solution:

S= {HH, HT, TH, TT} = 4

(i)     Pr (two tails) = {TT} = ¼

(ii)  Pr (=one head one tail) = {HT, TH} = 2/4 = 1/2

(iii)               Pr (two heads) = {HH} = ¼

(iv) Pr (1 head 1 tail in that order) = (HT) = 1/4

Die: A fair die is a six faced die. A die could be tossed in a different number of trials.

When a die is tossed once: {1, 2, 3, 4, 5, 6} = 6

When a die is tossed twice or 2 dice tossed, the outcome represented in the table below and the total outcome is 36.

(i)     (Total score of 10) = (4,6), (5,5), (6,4) = 3

Pr (total score of 10) = 3/36 = 1/12

(ii)  (at least 4) = (1,3),(1,4),(1,5),(1,6),(2,2),(2,3)……… = 33

Pr(total of at least 4) = 33/36 = 11/12

(iii)   (total score of a prime number) =

(1,1), (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,6), (6,1), (6,5), =15

Pr(total score of a prime number) = 15/36 =5/12

EVALUATION:

1.      Two fair dice are tossed. Find the probability (a) of not getting a total of 9 (iii) that the two dice show the same number.

2.      In a single throw of three fair coins, find the probability that: (a) one head two tails appear (b) at least one head appears.

MUTUALLY EXCLUSIVE EVENTS: Two events are mutually exclusive if they cannot occur at the same time. That is no common element between them. This leads to the ADDITION RULE.

Addition rule: :If E1, E2, E3, ….are dependent events than Pr(EUE2…. UEn) = Pr (E1) + Pr (E) + …. Pr(En)

Words such as; or, either are used to indicate addition of Probabilities.

Example: In a single throw of a fair die, what is the probability that an even number or a perfect square greater than 1 shows up?

Solution: S = {1, 2, 3, 4, 5, 6} = 6

Even number = {2, 4, 6} = 3, perfect square > 1 = {4}

Pr (even nos) = 3/6                 Pr (Perfect square> 1) = 1/6

Pr (even of perfect square) = 3/6 + 1/6

= 2/3