Probability

Question

Four red cards and four green cards are well shuffled. Cards are drawn, without replacement, until the color of a card drawn matches the color of the first card drawn. Find the average number of cards drawn until a match occurs.
The average number of cards drawn is (blank).

Answer 1

Let R symbolize drawing a red card and let G symbolize drawing a green card.

 

 

 

You might draw a red card first, in which case the possible outcomes until a matching red card is drawn are:

 

RR        2 total cards

RGR        3 total cards

RGGR        4 total cards

RGGGR        5 total cards

RGGGGR        6 total cards

 

 

 

Or, you might draw a green card first, in which case the possible outcomes until a matching green card is drawn are:

 

GG        2 total cards

GRG        3 total cards

GRRG        4 total cards

GRRRG        5 total cards

GRRRRG        6 total cards

 

 

 

That's a total of 10 outcomes. Add the total cards drawn for each outcome together and divide by 10 to find the average number of cards drawn:

 

2+2+3+3+4+4+5+5+6+6 = 4+6+8+10+12 = 10+18+12 = 10+30 = 40

40/10 = 4 cards drawn, on average