To solve this problem, we need to determine the number of derangements of the presidents' terms, as well as the total number of possible permutations.

A derangement is a permutation of a set in which no element appears in its original position. In this case, it means that no president should be matched with their term in office.

Let's denote the total number of presidents (24) as n.

The number of derangements for n is given by the following recurrence relation:

D(n) = (n-1)(D(n-1) + D(n-2))

with initial conditions D(1) = 0 and D(2) = 1.

Using this recurrence relation, we can calculate D(24):

D(3) = (3-1)(D(2) + D(1)) = (2)(1 + 0) = 2
D(4) = (4-1)(D(3) + D(2)) = (3)(2 + 1) = 9
D(5) = (5-1)(D(4) + D(3)) = (4)(9 + 2) = 44
D(6) = (6-1)(D(5) + D(4)) = (5)(44 + 9) = 265
D(7) = (7-1)(D(6) + D(5)) = (6)(265 + 44) = 1854
D(8) = (8-1)(D(7) + D(6)) = (7)(1854 + 265) = 14833
D(9) = (9-1)(D(8) + D(7)) = (8)(14833 + 1854) = 133496
D(10) = (10-1)(D(9) + D(8)) = (9)(133496 + 14833) = 1334961
D(11) = (11-1)(D(10) + D(9)) = (10)(1334961 + 133496) = 14684570
D(12) = (12-1)(D(11) + D(10)) = (11)(14684570 + 1334961) = 161730541
D(13) = (13-1)(D(12) + D(11)) = (12)(161730541 + 14684570) = 1775552102
D(14) = (14-1)(D(13) + D(12)) = (13)(1775552102 + 161730541) = 19469394261
D(15) = (15-1)(D(14) + D(13)) = (14)(19469394261 + 1775552102) = 213237862152
D(16) = (16-1)(D(15) + D(14)) = (15)(213237862152 + 19469394261) = 2335530910401
D(17) = (17-1)(D(16) + D(15)) = (16)(2335530910401 + 213237862152) = 25600888296580
D(18) = (18-1)(D(17) + D(16)) = (17)(25600888296580 + 2335530910401) = 281140877257601
D(19) = (19-1)(D(18) + D(17)) = (18)(281140877257601 + 25600888296580) = 3095769953579882
D(20) = (20-1)(D(19) + D(18)) = (19)(3095769953579882 + 281140877257601) = 34113428869167083
D(21) = (21-1)(D(20) + D(19)) = (20)(34113428869167083 + 3095769953579882) = 375165429561837450
D(22) = (22-1)(D(21) + D(20)) = (21)(375165429561837450 + 34113428869167083) = 4129676631751178811
D(23) = (23-1)(D(22) + D(21)) = (22)(4129676631751178811 + 375165429561837450) = 45493251749266976864
D(24) = (24-1)(D(23) + D(22)) = (23)(45493251749266976864 + 4129676631751178811) = 500644087983736878725

Now, let's calculate the total number of possible permutations:

P(n) = n!

P(24) = 24! = 6,204,484,903,868,800

Finally, the probability that no president is matched with his term in office is:

Probability = D(24) / P(24) = 500,644,087,983,736,878,725 / 6,204,484,903,868,800

Hence, the probability is approximately 0.0807, or 8.07%.