12. From a box containing a set of playing cards, the many ways to take one ace on the first take and one black card on the second take without being returned are ... A.100,B.101,C.102 ,D.103,E.104

Question

It has odd probability. I'm so confused sir. Can you help me?

Answer 1

A standard deck of cards has 52 cards, four of those are Aces, and 26 are black. Of the four Aces, two of those are black.

There are four ways to pick an Ace on the first draw.

Now, there are two possible scenarios for the second draw, depending on whether the Aces was red or black.

If the Ace was red, the 26 of the 51 remaining cards are black, so there would be 26 ways to get a black card with the ace being red. Since there are two red Aces, that gives 2 x 26 = 52 possibilities

But if the Ace was black (also two cards), then only 25 black cards remain so there would be 25 ways to get a black card on the second draw. That gives 2 x 25 = 50 possibilities.

Now, add the two results together to get the total number of possibilities, or 50 52 = 102. So the answer is C.

Answer 2

Although related to probability - this is a counting technique problem. You can tell by how the question starts "what is the number of way..." as opposed to "what is the probability..."

Before you begin, you must be familiar with a standard deck of cards - there are 52 cards. There are 4 aces (red heart, red diamond, black spade, black club) and there are also 26 black cards. Note that there are 2 cards that are BOTH an ace and black.

Since you have 2 draws - think about taking one card at a time:

1st draw - there are 4 aces - two are red and two are black

If the first card is a red ace, there are 26 black cards that can be taken. Since there are 2 red aces, that is a total of 2 x 26 = 52 so far. [NOTE: when you take the 2nd card, you have knowledge of what the 1st card is - this is known as conditional probability. The 2nd card is black, given the 1st card is a red ace]

OR

if the first card is a black ace, there are 25 black cards left that can be taken, there there are 2 black aces, that is a total of 2 x 25 = 50. [Since there is no replacement, you have 1 less black card available in the box because you have knowledge that the 1st card was a black ace]

Notice that I used the word "OR". These two scenarios are mutual exclusive events (cannot happen at the same time since you first pick a red ace or a black ace), therefore they can be added.

Ans: 52 + 50 = 102 (C)

You can also start a tree by choosing an ace first, then seeing how many black cards are left. You would then start with the 4 branches; one for each ace.