Question

A card is drawn at random from a deck of fifty-two cards. what is the probability of drawing a diamond, a card with an even number on it, or a card with a number divisible by three on it, but not a card that falls into more than one of these categories

Answer 1

Cards that fall into more than one of these categories:

2 of diamond, 4 of diamond, 6 of diamond, 8 of diamond, 10 of diamond(both diamond and even number)

3 of diamond, 6 of diamond, 9 of diamond(both diamond and divisible by three)

6 of clubs, 6 of spades, 6 of hearts, 6 of diamond(both even and divisible by three)

Cards that fall into only one of these categories:

1 of diamond, 5 of diamond, 7 of diamond, jack of diamond, queen of diamond, king of diamond(diamonds that aren't even or divisible by three)

Count: 6

2 of clubs, 4 of clubs, 8 of clubs, 10 of clubs, 2 of spades, 4 of spades, 8 of spades, 10 of spades, 2 of hearts, 4 of hearts, 8 of hearts, 10 of hearts(even number cards that aren't diamonds or divisible by three)

Count: 12

3 of clubs, 9 of clubs, 3 of spades, 9 of spades, 3 of hearts, 9 of hearts(cards divisible by three that aren't diamonds or even)

Count: 6

Total: 24

So the probability of drawing a diamond, a card with an even number on it, or a card with a number divisible by three on it, but not a card that falls into more than one of these categories is 24/52 or 6/13.

Answer 2

The probability of a diamond is 13/52 = 1/4 So none of the diamonds can be used for the other 2 restrictions.

The even numbered cards are 2 4  8 10.
The 6 cannot be used, because it could come into the next category. Six is divisible by 3. 
These can come from clubs hearts and spades. 
So each even number has 3 suits it can come from
The number of cards in this category is 3 * 4 = 12
The probability of this occurring is 12/39 = 4/13

The cards divisible by 3 are
3 and 9 [Again you can't use the 6]
The Total number of cards are 2*3 = 6
The probability is 6/39 = 2/13

The total probability (without drawing a 6) is
1/4 + 2/13 + 4/13 = 37/52

The two categories where a 6 can turn up are equally likely so choosing the category is 1/2. the probability of a 6 turning up in the first category is 1/2* 3/39 = 3/78 = 1/26

The probability of the 6 turning up in the second category is 1/2 * 3/39 = 3/78 = 1/26 So these two numbers must added into the mix.

37/52 + 2(1/26) = 37/52 + 1/13 = 41/52