sets word problem

Question

A card is drawn from a deck of 52 playing cards. Find the number of outcomes in which the card is either a queen or a black card.

Answer 1

This word problem on probability can trick you if you aren't careful.

 

The problem asks for the probability of EITHER a queen OR a black card. These two events are independent events, meaning one can happen, without the other. I can pick a black card that is not a queen. However, it IS possible to pick a black card that is a queen. The situation of a black card AND a queen being pulled means we have to adjust our probability formula.

 

The formula for the probability of two events when they are independent is:

 

P(Q or B) = P(Q) + P(B) - P(Q AND B)

 

P(Q AND B) = P(Q) x P(B)

 

Therefore

 

P(Q or B) = P(Q) + P(B) - (P(Q) x P(B))

 

P(Q or B) = 4/52 + 26/52 - (4/52 x 26/52)

 

P(Q or B) = 4/52 + 26/52 - (2/52)

 

P(Q or B) = 28/52

 

P(Q or B) = 7/13

 

 

Answer 2

This is an interesting question.

 

In order to get the answer, we need to consider three different situations:

 

1.Number of outcomes in which the card is a queen: 4

 

2.Number of outcomes in which the card is a black card: 26 (half of the cards are black)

 

3.Number of outcomes in which the card is both a queen and a black card:2 (half of the queen cards are black)

 

           (Notice: this is different from what is described in the question. The reason why we pay attention to this is that the black queens are counted two times in the first two situations.One time as the queen cards. Another time as the black cards.)

 

Thus, the number of outcomes in which the card is either a queen or a black card is:

 

     4 + 26 - 2 = 28

 

Hope this answer can help you :)

Answer 3

there are 4 queens out of 52 cards, so there are 4 outcomes for a queen

there are 26 black cards out of 52 cards so there are 26 outcomes for a black card

however, the outcomes overlap because there are 2 queens that are black so you don't include them twice

therefore the number of outcomes is 4+26-2=28 outcomes

looking at it another way...

there are 26 black cards, two of which are queens

that leaves only 2 (red) queens left

the number of outcomes is 26+2=28 (P=28/52=7/13)