Odds in favor/favour of an event - Odds against an event

Odds in favor/favour of an event

Odds in favour/favor of an event is the ratio of "Number of favorable choices (successes)" to "Number of unfavourable choices (failures)".

⇒ Odds in favor of an event = Number of favourable choices : Number of unfavorable choices
= m : mc
(Or) = Number of successes : Number of failures
= m : (n - m)

Examples

  1. In the experiment of tossing a coin,

    Total number of possible choices = 2 {HEAD, TAIL}
    ⇒ n = 2

    Let

    1. K : The event of getting a tail.

    • For Event K

    Number of favorable/favourable Choices
    (Or) Number of Successes

    =

    1 {TAIL}
    ⇒ mK = 1
    All elementary events of the experiment where we get a "TAIL" are favorable to event 'K'.

    Number of unfavorable/unfavourable Choices (Failures)
    = Total Number of possible choices
        - Number of Favorable/Favourable choices (successes)
    ⇒ mKc = n - mK
    = 2 - 1
    = 1

    Therefore,

    Odds in favor/favour = Number of favorable/favourable choices
    : Number of unfavorable/unfavourable choices
    (Or) = Number of Successes : Number of Failures
    = mK : mKc
    = 1 : 1

  2. In the experiment of throwing a die,

    Total number of possible choices = 6 {1, 2, 3, 4, 5, 6}
    ⇒ n = 6

    Let

    1. H : The event of getting getting a number greater than 4.

    • For Event H

    Number of favorable/favourable Choices
    (Or) Number of Successes

    =

    2 {FIVE, SIX}
    ⇒ mH = 2
    All elementary events of the experiment where we get a number greater than four are favorable to event 'H'.

    Number of unfavorable/unfavourable Choices (Failures)
    = Total Number of possible choices
        - Number of Favorable/Favourable choices (successes)
    ⇒ mHc = n - mH
    = 6 - 2
    = 4

    Therefore,

    Odds in favor/favour = Number of favorable/favourable choices
    : Number of unfavorable/unfavourable choices
    (Or) = Number of Successes : Number of Failures
    = mH : mHc
    = 2 : 4
    = 1 : 2

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Odds against an event

Odds against an Event is the ratio of "Number of unfavourable choices (failures)" to "Number of favorable choices (successes)".

⇒ Odds against an event = Number of unfavourable choices : Number of favorable choices
= mc : m
(Or) = Number of failures : Number of successes
= (n - m) : m

Example

  1. In the experiment of tossing a coin,

    Total number of possible choices = 2 {HEAD, TAIL}
    ⇒ n = 2

    Let

    1. G : The event of getting a tail.

    • For Event G

    Number of favorable/favourable Choices
    (Or) Number of Successes

    =

    1 {TAIL}
    ⇒ mG = 1
    All elementary events of the experiment where we get a "TAIL" are favorable to event 'G'.

    Number of unfavorable/unfavourable Choices
    (Or) Number of Failures

    =

    Total No. of possible choices
    - No. of Favourable choices (Successes).
    ⇒ mGc = n - mG
    = 2 - 1
    = 1

    Therefore,

    Odds against = No. of unfavourable choices : No. of favorable choices
    (Or) = No. of failures : No. of successes
    = mGc : mG
    = 1 : 1

  2. In the experiment of throwing a die,

    Total number of possible choices = 6 {1, 2, 3, 4, 5, 6}
    ⇒ n = 6

    Let

    1. T : The event of getting getting a number less than 6.

    • For Event T

    Number of favorable/favourable Choices
    (Or) Number of Successes

    =

    5 {ONE, TWO, THREE, FOUR, FIVE}
    ⇒ mT = 5
    All elementary events of the experiment where we get a number less than six are favorable to event 'T'.

    Number of unfavorable/unfavourable Choices
    (Or) Number of Failures

    =

    Total No. of possible choices
    - No. of Favourable choices (Successes).
    ⇒ mTc = n - mT
    = 6 - 5
    = 1

    Therefore,

    Odds against = No. of unfavourable choices : No. of favorable choices
    (Or) = No. of failures : No. of successes
    = mTc : mT
    = 1 : 5

Expressing odds for and against

Odds in favor/favour of as well as odds against are ratios expressing the relationship between the number of favorable choices or successes and number of unfavorable choices or failures. Reversing the antecedent (first term) and consequent (second term) of the ratio representing the odds in favor of would give the odds against and vice-versa.

Odds being a ratio can be expressed in all the different forms in which a mathematical ratio can be expressed.

Odds as a fraction

The ratio a : b can be expressed as
a
b

Odds in a decimal form

odds in favor and odds against can also be expressed in a decimal form

Examples

  1. In the experiment of throwing a die,

    Total number of possible choices = 6 {1, 2, 3, 4, 5, 6}
    ⇒ n = 6

    Let

    1. F : The event of getting a number not divisible by 3

    • For Event F

    Number of favorable/favourable Choices
    (Or) Number of Successes

    =

    4 {ONE, TWO, FOUR, FIVE}
    ⇒ mF = 5
    All elementary events of the experiment where we get a number not divisible by three are favorable to event 'F'.

    Number of unfavorable/unfavourable Choices = Total No. of possible choices
    - No. of Favourable choices.
    (Or) Number of Failures = Total No. of possible choices
    - No. of Successes.
    ⇒ mFc = n - mF
    = 6 - 4
    = 2

    Therefore,

    Odds in favor = No. of favourable choices : No. of unfavorable choices
    (Or) = No. of successes : No. of failures
    = mF : mFc
    = 4 : 2
    = 2 : 1

    Odds against = No. of unfavourable choices : No. of favorable choices
    (Or) = No. of failures : No. of successes
    = mFc : mF
    = 2 : 4
    = 1 : 2

    • Odds as a Fraction
    odds in favor =
    mF
    mFc
    =
    4
    2
    = 2
    odds against =
    mFc
    mF
    =
    2
    4
    =
    1
    2
    • Odds in decimal form
    odds in favor =
    mF
    mFc
    =
    4
    2
    = 2.0
    odds against =
    mFc
    mF
    =
    2
    4
    =
    1
    2
    = 0.5

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