Probability question

Question

40 cards are in a deck from Ace to 10 and a there is a die with #s 1-6 on it. You must roll that die and then draw two cards from the deck, one at a time without replacing the first. Find the probability that sum of the cards and the die is 8

Answer 1

The easiest way to get the answer is to see this as a series of events. We roll the die, with 6 possibilities. Then we pick two cards without replacement. For the cards, the value of the first card and the second is equal to 8 - value of the die. Consider the possibilities:

Die = 1 so cards = 7, or 1 and 6, 2 and 5, 3 and 4. Doesn't matter which comes first, so in total this is 6 ways. 

Die = 2 so cards = 6, or 1 and 5, 2 and 4, 3 and 3 and again 6 ways.

 

I'll let you finish the counting of possibilities. Once you have the total number of ways you can get 8, you divide by the total number of possibilities given any sum. To help with that, consider that there are 40 cards for the first pick (with ten different values) and 39 ways to pick the second. Think of 10 chances for 1, 10 chances for 2, etc. Once you pick one, what are the possible number of sums you can get?

 

I hope I gave you enough to solve this. I hesitate to provide the solution, as you now can get the probability yourself. If you were to estimate it, you could average the die at 3, and the cards at a total of 5, so only less than 1/3 of the cards can work, and about half of the die rolls. So that would be about 1/6th of the total. Total picks of cards is 40*39. 

 

Nice problem - hope you try to finish it.