What is the probability of pulling 2 specific cards in a hand of 8 from 78 cards?

Question

I drew a hand that contains the Sun and the Moon from a tarot deck with $78$ cards. I want to find the probability of this occurring. I think because there are $78 \choose 6$ distinct arrangements of the $6$ other cards in the hand, and $78\choose8$ distinct hands, the probability of choosing both the Sun and the Moon is $$\frac{78 \choose 6}{78 \choose 8}=1/9.$$ But $1/9$ seems pretty high. Am I missing something?

Answer 1

It would be

$$\frac{1\cdot{76\choose 6}}{{78\choose 8}}\approx0.009324009$$

where you can replace $1$ with ${2\choose 2}$

Answer 2

Simulation (to check):

deck = c(1,1,rep(0,76))
table(deck)
deck
 0  1 
76  2 

set.seed(406)
sm = replicate(10^6, sum(sample(deck,8)))
mean(sm==2)
[1] 0.009219       # aprx 0.0093
sd(sm==2)/1000
[1] 9.557206e-05   # 95% marg of sim err

Simulated value is $0.0092 \pm 0.0001.$

choose(76, 6)/choose(78,8)
[1] 0.009324009