probability calculation on excel

Question

I got an excel sheet which shows a life table as below, note that q(x) is the probability of dying between age x and x+1. I am now focus on the probability of death after age 65, and I want to know the probability of a person who alive at age 65 and dies at age 100 and a person alive at 65 and dies at age 85?

    Probability
    of dying
    between
    ages x to x+1
Age        q(x)
0-1     0.006667
1-2     0.000449
2-3     0.000322
3-4     0.000247
4-5     0.000178
5-6     0.000166
6-7     0.000147
7-8     0.000129
8-9     0.000109
9-10    0.000087
10-11   0.000072
11-12   0.000078
12-13   0.000121
13-14   0.000209
14-15   0.000328
15-16   0.000451
16-17   0.000569
17-18   0.000690
18-19   0.000817
19-20   0.000945
20-21   0.001084
21-22   0.001216
22-23   0.001311
23-24   0.001354
24-25   0.001358
25-26   0.001348
26-27   0.001344
27-28   0.001345
28-29   0.001359
29-30   0.001384
30-31   0.001414
31-32   0.001444
32-33   0.001475
33-34   0.001506
34-35   0.001542
35-36   0.001592
36-37   0.001659
37-38   0.001738
38-39   0.001830
39-40   0.001941
40-41   0.002064
41-42   0.002217
42-43   0.002421
43-44   0.002684
44-45   0.002987
45-46   0.003303
46-47   0.003624
47-48   0.003968
48-49   0.004342
49-50   0.004746
50-51   0.005172
51-52   0.005617
52-53   0.006093
53-54   0.006611
54-55   0.007174
56-57   0.008451
57-58   0.009121
58-59   0.009775
59-60   0.010415
60-61   0.011075
61-62   0.011791
62-63   0.012577
63-64   0.013484
64-65   0.014542
65-66   0.015783
66-67   0.017195
67-68   0.018699
68-69   0.020247
69-70   0.021917
70-71   0.023725
71-72   0.025734
72-73   0.028077
73-74   0.030750
74-75   0.033815
75-76   0.037090
76-77   0.040540
77-78   0.044677
78-79   0.049227
79-80   0.054348
80-81   0.060110
81-82   0.066576
82-83   0.073449
83-84   0.080709
84-85   0.090777
85-86   0.101080
86-87   0.112324
87-88   0.124544
88-89   0.137762
89-90   0.151991
90-91   0.167224
91-92   0.183440
92-93   0.200596
93-94   0.218632
94-95   0.237462
95-96   0.256985
96-97   0.277076
97-98   0.297597
98-99   0.318395
99-100  0.339311
100+    1.000000

Answer 1

The answer is simple (assuming the data is in A1:B101 without headers / start age is at F1 / end age is at F2):

=1-PRODUCT(1-INDEX(B:B,F1+1):INDEX(B:B,F2))

This is an array formula and must be confirmed with ctrl + shift + enter !

Note: The formula will fail for ages >55 because there is no 55-56 range!

Answer 2

Set random variables X[i], it equals to 1 when one person is alive at the age of i, otherwise it equals to 0. Then using conditional probability, q(x) = P{X[x]=0|X[x-1]=1...X[1]=1,X[0]=1}

What you want to calculate is P{X[85]=0,X[84]=1...X[66]=1|X[65]=1,X[64]=1...X[0]=1} , set it as ANS.

So,try this :

ANS = P{X[85]=0,X[84]=1...X[66]=1|X[65]=1,X[64]=1...X[0]=1}
(using probabilistic decomposition) = 
P{X[85]=0|X[84]=1,X[83]=1...X[0]=1} * P{X[84]=1|X[83]=1,X[82]=1...X[0]=1}
* P{X[83]=1|X[82]=1,X[81]=1...X[0]=1} * P{X[82]=1|X[81]=1,X[80]=1...X[0]=1}
* ... * P{X[66]=1|X[65]=1,X[64]=1...X[0]=1}
=q(85)[1-q(84)][1-q(83)]...[1-q(66)]