Provide Information Regarding Statistics & Econometrics : Classical Definition of Probability Lecture 18

Classical Definition of Probability

Lecture 18

If a random experiment produces n equally likely and mutually exclusive outcomes (referred to as total possible outcomes) and if an event is observed to occur m times (referred to as favorable), then the probability of the event is equal to the ratio of favorable to the total outcomes.

That’s S = {1, 2, 3,..., n} and A = {1, 2,..., m}

Where m < n

The number of outcomes in sample space is denoted by n(S) = n and the number of outcomes in event A is denoted by n(A) = m

The probability of event A can be obtained as:

Where P(A) lies between 0 and 1.

Examples based Coin

Example 5.3: Two coins are tossed once (OR a coin is tossed two times). Find the probability of

i. At least one head.

ii. Exactly one head.

iii. No head.

Solution: The sample space of a coin tossed twice or two coins are tossed once is given by;

S = {HH, HT, TH, TT}

n(S) = 4

i. Let A represent at least one head.

A = {HH, HT, TH}

n(A) = 3

Where P(A) lies between 0 and 1.

The probability of at least one head is given by:

ii. Let B represent at exactly one head.

B = {HT, TH}

n(B) = 2

The probability of exactly one head is given by:

iii. Let C represent no head.

C = {TT}

n (C) = 1

The probability of no head is given by:

Example 5.4: Two coins are tossed once (OR a coin is tossed two times). Find the probability of

i. At least one head.

ii. Exactly one head.

iii. Exactly two heads

iv. No head.

Solution: The sample space consists of 8 sample points listed below:

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

n(S) = 8

i. Let A represent at least one head.

A= {HHH, HHT, HTH, THH, HTT, THT, TTH}

n(A) = 7

ii. Let B represent exactly one head.

B = {HTT, THT, TTH}

n(B) = 3

iii. Let C represent exactly two heads.

C = {HHT, HTH, THH}

n(C) = 3

iv. Let D represent no head.

D = {TTT}

n(D) = 1

Examples-based Dice Experiments

Example 5.5: Two dice are rolled once. Find the probability

i. Same outcomes on both dies.

ii. The sum of dots is 6.

iii. At least one 5 on either die.

iv. The sum is less than 4.

Solution: The sample space consists of 36 outcomes, tabulated below:

i. Let A represent the same outcomes

A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

n (A) = 6

ii. Let B represent the sum of 6.

B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}

n (B) = 5

iii. Let C represent at least one 5 on either die.

C = {(1, 5), (2, 5), (3, 5), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 5)}

n (C) = 11

iv. Let D represent the sum is less than 4.

D = {(1, 1), (1, 2), (2, 1)}

n (D) = 3

Example based on Cards Experiments

Illustration of Standard Playing Cards Suit

A standard playing card deck consists of 52 playing cards; these 52 cards are divided into 4 suites of diamonds, hearts, spades, and clubs. Each suit contains 13 cards, i.e., Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.

Example 5.6: A card is selected from a standard deck of playing cards. Find the probability that the selected card is

i. Red

ii. Diamond

iii. Ace

iv. Face

v. King of diamond

Solution: A standard deck consists of 52 playing cards.

i. Let A represent the selected card is red

ii. Let B represent the selected card as a diamond

iii. Let C represent the selected card, which is Ace

iv. Let D represent the selected card's face

v. Let E represent king of diamond

Example 5.7: A bag contains 4 red and 3 white balls. Two balls are selected at random. Find the probability that

i. Both balls are red

ii. Alternate of colors

iii. At least one is red

Both balls are white

Solution: Summary of information:

Red balls = 4, White balls = 3, Total balls = 7, Selected balls = 2

The sample in this kind of experiment is obtained by combination technique, given below:

i. Let A represent both balls are red.

ii. Let B represent alternate colors.

iii. Let C represent that at least one is red (one or more than one is red)

Example 5.8: An urn contains 4 red, 6 black, and 3 white balls. Three balls are selected at random, find the probability that the selected balls are;

i. 1 red ball

ii. 2 black balls

iii. At least one white ball

Solution: Summary of information:

Red balls = 4, White balls = 6, White balls = 3, Total balls = 13, Selected balls = 3

The sample in this kind of experiment is obtained by combination technique, given below: