Chapter 4, Problem 5CS

To determine

Compute the probability that person alive at ages 50, 60,70,80,90, and 100.

Answer to Problem 5CS

The probabilities are,

$\begin{array}{l}P\left(\text{alive at age 50}\right)=0.93782\\ P\left(\text{alive at age 60}\right)=0.88164\\ P\left(\text{alive at age 70}\right)=0.76666\\ P\left(\text{alive at age 80}\right)=0.54385\\ P\left(\text{alive at age 90}\right)=0.22931\\ P\left(\text{alive at age 100}\right)=0.02809\end{array}$.

Explanation of Solution

Calculation:

The given information is a life table. The first and second column of the table represents the age groups and corresponding proportion of survivals. The first value in the proportion column gives the proportion people who survived between 0 to 10. The second value in the proportion column gives the conditional probability that person is alive at the end of the age interval given that they were alive at the beginning of the age interval.

The formula for conditional probability is given by:

$P\left(B|A\right)=\frac{P\left(A\text{and}B\right)}{P\left(A\right)}$.

From the conditional probability,

$P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B|A\right)$

Alive at the age 50:

$P\left(\text{alive at age 50}\right)=P\left(\text{alive at age 40}\right)\times P\left(\text{alive at age 50}|\text{alive at age 40}\right)$

From Exercise 4, $P\left(\text{alive at age 40}\right)=0.96533$.

From table,

$\begin{array}{c}P\left(\text{alive at age 50|}\text{alive at age 40}\right)=P\left(\text{alive at 40-50}\right)\\ =0.97150\end{array}$.

Substitute these values in the above formula.

Therefore,

$\begin{array}{c}P\left(\text{alive at age 50}\right)=\left(0.96533\right)\times \left(0.97150\right)\\ =0.937818\\ =0.93782\end{array}$

Thus, $P\left(\text{alive at age 50}\right)=0.93782$.

Alive at the age 60:

$P\left(\text{alive at age 60}\right)=P\left(\text{alive at age 50}\right)\times P\left(\text{alive at age 60}|\text{alive at age 50}\right)$

From the above, $P\left(\text{alive at age 50}\right)=0.93782$.

From table,

$\begin{array}{c}P\left(\text{alive at age 60|}\text{alive at age 50}\right)=P\left(\text{alive at 50-60}\right)\\ =0.94010\end{array}$.

Substitute these values in the above formula,

Therefore,

$\begin{array}{c}P\left(\text{alive at age 60}\right)=\left(0.93782\right)\times \left(0.94010\right)\\ =0.88164\end{array}$

Thus, $P\left(\text{alive at age 60}\right)=0.88164$.

Alive at the age 70:

$P\left(\text{alive at age 70}\right)=P\left(\text{alive at age 60}\right)\times P\left(\text{alive at age 70}|\text{alive at age 60}\right)$

From the above, $P\left(\text{alive at age 60}\right)=0.88164$.

From table,

$\begin{array}{c}P\left(\text{alive at age 70|}\text{alive at age 60}\right)=P\left(\text{alive at 60-70}\right)\\ =0.86958\end{array}$.

Substitute these values in the above formula.

Therefore,

$\begin{array}{c}P\left(\text{alive at age 70}\right)=\left(0.88164\right)\times \left(0.86958\right)\\ =0.766656\\ =0.76666\end{array}$

Thus, $P\left(\text{alive at age 70}\right)=0.76666$.

Alive at the age 80:

$P\left(\text{alive at age 80}\right)=P\left(\text{alive at age 70}\right)\times P\left(\text{alive at age 80}|\text{alive at age 70}\right)$

From the above, $P\left(\text{alive at age 70}\right)=0.76666$.

From table,

$\begin{array}{c}P\left(\text{alive at age 80|}\text{alive at age 70}\right)=P\left(\text{alive at 70-80}\right)\\ =0.70938\end{array}$.

Substitute these values in the above formula.

Therefore,

$\begin{array}{c}P\left(\text{alive at age 80}\right)=\left(0.76666\right)\times \left(0.70938\right)\\ =0.54385\end{array}$

Thus, $P\left(\text{alive at age 80}\right)=0.54385$.

Alive at the age 90:

$P\left(\text{alive at age 90}\right)=P\left(\text{alive at age 80}\right)\times P\left(\text{alive at age 90}|\text{alive at age 80}\right)$

From the above, $P\left(\text{alive at age 80}\right)=0.54385$.

From table,

$\begin{array}{c}P\left(\text{alive at age 90|}\text{alive at age 80}\right)=P\left(\text{alive at 80-90}\right)\\ =0.42164\end{array}$.

Substitute these values in the above formula.

Therefore,

$\begin{array}{c}P\left(\text{alive at age 90}\right)=\left(0.54385\right)\times \left(0.42164\right)\\ =0.229308\\ =0.22931\end{array}$

Thus, $P\left(\text{alive at age 90}\right)=0.22931$.

Alive at the age 100:

$P\left(\text{alive at age 100}\right)=P\left(\text{alive at age 90}\right)\times P\left(\text{alive at age 100}|\text{alive at age 90}\right)$

From the above, $P\left(\text{alive at age 90}\right)=0.22931$.

From table,

$\begin{array}{c}P\left(\text{alive at age 100|}\text{alive at age 90}\right)=P\left(\text{alive at 90-100}\right)\\ =0.12248\end{array}$.

Substitute these values in the above formula.

Therefore,

$\begin{array}{c}P\left(\text{alive at age 100}\right)=\left(0.22931\right)\times \left(0.12248\right)\\ =0.028086\\ =0.02809\end{array}$

Thus, $P\left(\text{alive at age 100}\right)=0.02809$.