The probability of surviving is given as 20%.
It is required to find the odds of surviving.
Let the event "surviving a certain type of cancer" be X.
It follows that the probability can be written as:
[tex]\begin{gathered} P(X)=20\%=\frac{20}{100}=\frac{\cancel{20}^1}{\cancel{100}^5} \\ \Rightarrow P(X)=\frac{1}{5} \end{gathered}[/tex]Recall that for an event X, the odds in favor of the event happening are given by the ratio:
[tex]\begin{gathered} \frac{P(X)}{P(\overline{X})} \\ \text{Where }P(\overline{X})\text{ is the probability of the event not happening} \end{gathered}[/tex]The probability of event X not happening is given as:
[tex]\begin{gathered} P(\overline{X})=1-P(X) \\ \text{Substitute }P(X)=\frac{1}{5}\text{ into the equation:} \\ P(\overline{X})=1-\frac{1}{5}=\frac{4}{5} \\ \Rightarrow P(\overline{X})=\frac{4}{5} \end{gathered}[/tex]Substitute the probabilities into the odds formula:
[tex]\frac{P\mleft(X\mright)}{P(\overline{X})}=\frac{\frac{1}{5}}{\frac{4}{5}}=\frac{1}{4}[/tex]The odds of surviving is 1:4.
