Chapter 6, Problem 1RQ

Summary Introduction

To determine: The probability of surviving for 3 years.

Introduction: Survival probability is the proportion of units that estimates the survival chances of people in a given length of time.

Explanation of Solution

Before calculating the probability of surviving for 3 years, it is required to calculate the probability of surviving for every in the contingency table. The probability (P) of surviving for 2012 can be calculated as follows:

$\begin{array}{c}{\text{P}}_{\text{2012}}\text{=}\frac{\text{The}\text{total}\text{number}\text{of}\text{survivors}}{\text{Total}\text{number}\text{of}\text{patients}\text{treated}\text{in}\text{three}\text{years}}\\ =\frac{152}{180}\\ =0.844\end{array}$

$\begin{array}{l}\text{For the year 2014, the total number of survivors = 87}\hfill \\ \text{A total number of survivors after the first anniversary =152}\hfill \end{array}$

The probability of surviving for 2014 can be calculated as follows:

$\begin{array}{c}{P}_{2014}=\left(\frac{87}{152-37}\right)\\ =\left(\frac{87}{115}\right)\\ =0.756\end{array}$

$\begin{array}{l}\text{In the year 2015, the total number of survivors = 48}\hfill \\ \text{A total number of survivors after the first anniversary =87}\hfill \end{array}$

The probability of surviving for 2015 can be calculated as follows:

$\begin{array}{c}{P}_{2015}=\left(\frac{48}{87-31}\right)\\ =\left(\frac{48}{56}\right)\\ =0.857\end{array}$

The probability of surviving for 3 years can be calculated from the product of 2012, 2014, and 2015:

$\begin{array}{c}{\text{P}}_{\text{2012}}{\text{× P}}_{\text{2014}}{\text{× P}}_{\text{2015}}\text{=}\text{0}\text{.844}\times \text{0}\text{.756}\times \text{0}\text{.857}\\ \text{= 0}\text{.549}\\ \text{= 54}\text{.9\%}\end{array}$

Conclusion

The probability of surviving for 3 years is 54.9%.