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

Answer:

(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.

Step-by-step explanation:

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

$\text{Odds For}=\frac{P(E)}{1-P(E)}\\\\\text{Odds Against}=\frac{1-P(E)}{P(E)}$

(a)

The probability of the event E is:

$P(E)=\frac{8}{9}$

Compute the odds for and against E as follows:

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

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

(b)

The probability of the event E is:

$P(E)=\frac{7}{9}$

Compute the odds for and against E as follows:

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

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:

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

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:

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

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