a)
[tex]\frac{2}{7}[/tex]
b)
[tex]\frac{12}{11}[/tex]
Explanation
The probability of an event occurring is intuitively understood to be the likelihood or chance of it occurring, the probability is givenby:
[tex]P(A)=\frac{favourable\text{ outcomes}}{total\text{ outcomes}}[/tex]
so
Step 1
we can find the odds in favor by using the expression:
[tex]Odds\text{ in favor=}\frac{P(A)}{1-P(A)}[/tex]
so
let
[tex]\begin{gathered} P(A)=unknown \\ odds\text{ in favor=}\frac{2}{5} \end{gathered}[/tex]
now, replace and solve for P(A)
[tex]\begin{gathered} Odds\text{ in favor=}\frac{P(A)}{1-P(A)} \\ \frac{2}{5}=\frac{P(A)}{1-P(A)} \\ cross\text{ multiply } \\ 2(1-P(A))=5*P(A) \\ 2-2P(A)=5P(A) \\ add\text{ 2P\lparen A\rparen in both sides} \\ 2-2P(A)+2P(A)=5P(A)+2P(A) \\ 2=7P(A) \\ divide\text{ both sides by 7} \\ \frac{2}{7}=\frac{7P(A)}{7} \\ \frac{2}{7}=P(A) \end{gathered}[/tex]
therefore, the probabilyt of winning a new TV is
[tex]\frac{2}{7}[/tex]
Step 2
now, to find the odds against we need to use the formula:
[tex]odds\text{ agains=}\frac{1-P(A)}{P(A)}[/tex]
so
let
[tex]P(B)=\frac{11}{23}[/tex]
now, replace in the formula and calculate
[tex]\begin{gathered} odds\text{ against=}\frac{1-P(A)}{P(A)} \\ odds\text{ against=}\frac{1-\frac{11}{23}}{\frac{11}{23}} \\ odds\text{ against=}\frac{\frac{12}{23}}{\frac{11}{23}}=\frac{12*23}{11*23}=\frac{12}{11} \\ odds\text{ against=}\frac{12}{11} \end{gathered}[/tex]
therefore, the odds against are
[tex]\frac{12}{11}[/tex]
I hope this helps you
