Chapter 5, Problem 3CS

To determine

The condition probability of $P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)$ .

Answer to Problem 3CS

The probability at age 30 $P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)$ is $0.97801$ .

Explanation of Solution

Given:

The life table is given as follows:

Concept used:

  $P\left(\text{Alive}\text{at}\text{age}x\text{/Alive}\text{age}y\right)$ is the conditional probability of alive person at the end x age interval, given that they were alive at the beginning of the age interval. Probability at x can express as follows:

  $P\left(\text{Alive}\text{at}\text{age}x\right)=P\left(\text{Alive}\text{at}\text{age}x\text{/Alive}\text{age}y\right)\times P\left(\text{Alive}\text{age}y\right)$

Calculation:

From the above table,

  $\begin{array}{l}P\left(\text{Alive}\text{at}\text{age}\text{20/Alive}\text{age}10\right)=0.99613\left(\text{proportion}\text{surviving}\text{in}\text{interv 10-20}\right)\\ P\left(\text{Alive}\text{at}\text{age}\text{20/Alive}\text{age}10\right)=0.99613\end{array}$

The probability $P\left(\text{Alive}\text{at}\text{age}\text{20}\right)$ is calculated as follows:

  $\begin{array}{l}P\left(\text{Alive}\text{at}\text{age}\text{20}\right)=P\left(\text{Alive}\text{at}\text{age}\text{20/Alive}\text{age}10\right)\times P\left(\text{Alive}\text{age}10\right)\\ P\left(\text{Alive}\text{at}\text{age}\text{20}\right)=0.99613\times 0.99123\\ P\left(\text{Alive}\text{at}\text{age}\text{20}\right)=0.98739\end{array}$

From the above table,

  $\begin{array}{l}P\left(\text{Alive}\text{at}\text{age}3\text{0/Alive}\text{age}20\right)=0.99050\left(\text{proportion}\text{surviving}\text{in}\text{interv 20-30}\right)\\ P\left(\text{Alive}\text{at}\text{age}3\text{0/Alive}\text{age}20\right)=0.99050\end{array}$

The probability $P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)$ is calculated as follows:

  $\begin{array}{l}P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)=P\left(\text{Alive}\text{at}\text{age}3\text{0/Alive}\text{age}20\right)\times P\left(\text{Alive}\text{age}20\right)\\ P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)=0.99050\times 0.98739\\ P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)=0.97801\end{array}$

Thus, the probability at age 30 $P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)$ is $0.97801$ .

Conclusion:

The probability at age 30 $P\left(\text{Alive}\text{at}\text{age}3\text{0}\right)$ is $0.97801$ .