A Grandpa and a Dog

Background

My friend sent me this TikTok in which the OP calculated the probability that his grandpa, 84, and his dog, 15, died within two days of each other.

OP's Analysis

Grandpa was alive for D = 84 years = 30,660 days. Dog, d = 15 years = 5,475 days.

The probability that Grandpa died within any 2-day period, P = 2 / D. Probability for Dog, p = 2 / d.

The probability that Grandpa and Dog died within the same 2-day period = P ⋅ p = 1 in ~42 million.

Issues with OP's Analysis

1. "Within 2 days of each other" is a 4-day period, not 2 days.
2. The probability of either dying on a given day is not evenly distributed throughout their lives.
3. Dog was born when Grandpa was age 69, so the probability of both dying days apart is 0% up until Grandpa turns 69.

Vastly More Accurate Calculation

Answer

• The probability P that Grandpa and Dog died within 2 days of each other is 0.0453% or 1 in about 2,200. This is a factor of 19,000 more likely than OP suggests.
• If there are 1,530 similar Grandpa-Dog pairs (Grandpa is 69 years older than Dog), the probability of at least one pair of them dying within 2 days of each other is 50%. p = 1 - (1 - P)^N.

Assumptions

1. Grandpa was an average American. That's so that we can use this data from the SSA describing the probability of death at any given age in the year 2017.
2. To be conservative, Dog is smaller indicating Dog will live longer and reduce the probability in question. Dog's age can translate to human age using this table from WebMD.
3. The death probability of Dog in any given year is the linear average death probability for a human covering the same timeframe.
4. Grandpa dying and Dog dying are two independent events and are only dependent on age.

Calculation

1. The probability of dying over the next year D_t comes from the SSA table.
2. If we guarantee both die within a given year, the probability P of one dying within 2 days of the other is 4 / 365 = 1.1%.
3. The probability M_t one made it to a given age t is the probability they made it to the previous age multiplied by the probability they did not die in the last year, M_t-1 ⋅ [ 1 - D_{t -1}].
4. The probability Y_t one died at age t is the probability they made it to t multiplied by the probability they will die this year, M_t ⋅ D_t .
5. The probability p_t both died within 2 days of each other in a given year is P ⋅ Y_t ⋅ y_t .
6. The overall probability of Grandpa dying within 2 days of Dog is the summation of all p_t.

Table

Summary of probabilities starting with the year Dog was born (Grandpa was age 69).

Grandpa				Dog				Combined
Age	M	D	Y	Age	m	d	y	Y ⋅ y	p
69	100.0%	2.1%	2.1%	0	100.0%	0.0%	0.0%	0.00%	0.0000%
70	97.9%	2.3%	2.2%	1	100.0%	0.1%	0.1%	0.00%	0.0000%
71	95.6%	2.5%	2.4%	2	99.9%	0.2%	0.2%	0.00%	0.0000%
72	93.3%	2.7%	2.5%	3	99.7%	0.2%	0.2%	0.00%	0.0001%
73	90.7%	3.0%	2.7%	4	99.5%	0.2%	0.2%	0.01%	0.0001%
74	88.1%	3.2%	2.9%	5	99.3%	0.2%	0.2%	0.01%	0.0001%
75	85.2%	3.6%	3.0%	6	99.1%	0.3%	0.3%	0.01%	0.0001%
76	82.2%	3.9%	3.2%	7	98.8%	0.3%	0.3%	0.01%	0.0001%
77	78.9%	4.3%	3.4%	8	98.5%	0.5%	0.5%	0.02%	0.0002%
78	75.5%	4.8%	3.6%	9	98.0%	0.7%	0.7%	0.02%	0.0003%
79	71.9%	5.3%	3.8%	10	97.4%	1.0%	0.9%	0.04%	0.0004%
80	68.1%	5.8%	4.0%	11	96.4%	1.3%	1.2%	0.05%	0.0005%
81	64.1%	6.5%	4.1%	12	95.2%	1.7%	1.6%	0.07%	0.0007%
82	60.0%	7.2%	4.3%	13	93.6%	2.2%	2.1%	0.09%	0.0010%
83	55.7%	7.9%	4.4%	14	91.5%	3.1%	2.9%	0.13%	0.0014%
84	51.3%	8.8%	4.5%	15	88.7%	4.6%	4.1%	0.18%	0.0020%
85	46.7%	9.8%	4.6%	16	84.6%	6.8%	5.8%	0.27%	0.0029%
86	42.2%	10.9%	4.6%	17	78.8%	10.4%	8.2%	0.38%	0.0041%
87	37.6%	12.1%	4.5%	18	70.6%	15.8%	11.2%	0.51%	0.0056%
88	33.0%	13.4%	4.4%	19	59.5%	23.2%	13.8%	0.61%	0.0067%
89	28.6%	14.9%	4.3%	20	45.6%	31.1%	14.2%	0.61%	0.0067%
90	24.3%	16.5%	4.0%	21	31.4%	38.2%	12.0%	0.48%	0.0053%
91	20.3%	18.3%	3.7%	22	19.4%	46.4%	9.0%	0.33%	0.0037%
92	16.6%	20.2%	3.3%	23	10.4%	56.4%	5.9%	0.20%	0.0022%
93	13.2%	22.2%	2.9%	24	4.5%	68.5%	3.1%	0.09%	0.0010%
94	10.3%	24.3%	2.5%	25	1.4%	81.2%	1.2%	0.03%	0.0003%