The new deck has 60 cards, of which (assuming both decks are standard), 4 ≤ (# of kings) ≤ 6. (There are the four normal kings, to which she could have added one black king or two black kings.)
There are two red aces in the new deck, so the odds of picking a red ace are 2/60, or 1/30.
Let's call x the number of kings. We have
(Odds of Red Ace) + (Odds of King) > 1/8
(We can add these without adjustments, as they're mutually exclusive: when you pick a card it can't be both a red ace and a king.)
1/30 + x/60 > 1/8
(x+2)/60 > 1/8
x > 5.5, meaning we must have 6 kings.
So the deck has 60 cards, 6 of which are kings, 4 of which are black.
(Odds of Black King) + (Odds of Red Jack) = 4/60 + 2/60 = 1/15 + 1/30 = 1/10
The OA (as posted) is incorrect.
There are two red aces in the new deck, so the odds of picking a red ace are 2/60, or 1/30.
Let's call x the number of kings. We have
(Odds of Red Ace) + (Odds of King) > 1/8
(We can add these without adjustments, as they're mutually exclusive: when you pick a card it can't be both a red ace and a king.)
1/30 + x/60 > 1/8
(x+2)/60 > 1/8
x > 5.5, meaning we must have 6 kings.
So the deck has 60 cards, 6 of which are kings, 4 of which are black.
(Odds of Black King) + (Odds of Red Jack) = 4/60 + 2/60 = 1/15 + 1/30 = 1/10
The OA (as posted) is incorrect.
