Question

Given the following probabilities for an event​ E, find the odds for and against E. ​(A) eight ninths ​(B) seven ninths ​(C) 0.59 ​(D) 0.71

Answer 1

(a) The odds for and against E are (8:1) and (1:8) respectively.

(b) The odds for and against E are (7:2) and (2:7) respectively.

(c) The odds for and against E are (59:41) and (41:59) respectively.

(d) The odds for and against E are (71:29) and (29:71) respectively.

Explanation:

The formula for the odds for an events E and against and event E are:

`\text{Odds For}=(P(E))/(1-P(E))\\\\\text{Odds Against}=(1-P(E))/(P(E))`

(a)

The probability of the event E is:

`P(E)=(8)/(9)`

Compute the odds for and against E as follows:

`\text{Odds For}=(P(E))/(1-P(E))=(8/9)/(1-(8/9))=(8/9)/(1/9)=(8)/(1)\\\\\text{Odds Against}=(1-P(E))/(P(E))=(1-(8/9))/(8/9)=(1/9)/(8/9)=(1)/(8)`

Thus, the odds for and against E are (8:1) and (1:8) respectively.

(b)

The probability of the event E is:

`P(E)=(7)/(9)`

Compute the odds for and against E as follows:

`\text{Odds For}=(P(E))/(1-P(E))=(7/9)/(1-(7/9))=(7/9)/(2/9)=(7)/(2)\\\\\text{Odds Against}=(1-P(E))/(P(E))=(1-(7/9))/(7/9)=(2/9)/(7/9)=(2)/(7)`

Thus, the odds for and against E are (7:2) and (2:7) respectively.

(c)

The probability of the event E is:

`P(E)=0.59`

Compute the odds for and against E as follows:

`\text{Odds For}=(P(E))/(1-P(E))=(0.59)/(1-0.59)=(0.59)/(0.41)=(59)/(41)\\\\\text{Odds Against}=(1-P(E))/(P(E))=(1-0.59)/(0.59)=(0.41)/(0.59)=(41)/(59)`

Thus, the odds for and against E are (59:41) and (41:59) respectively.

(d)

The probability of the event E is:

`P(E)=0.71`

Compute the odds for and against E as follows:

`\text{Odds For}=(P(E))/(1-P(E))=(0.71)/(1-0.71)=(0.71)/(0.29)=(71)/(29)\\\\\text{Odds Against}=(1-P(E))/(P(E))=(1-0.71)/(0.71)=(0.29)/(0.71)=(29)/(71)`

Thus, the odds for and against E are (71:29) and (29:71) respectively.