Question

From an ordinary deck of playing cards, cards are drawn successively at random and without replacement. Compute the probability that the third spade appears on the sixth draw.

Short answer

The probability that the third spade will appear on the sixth draw from a standard playing deck is approximately 0.03498 or 3.498%.

Step-by-step solution

Determine the possible cases for favorable outcomes

In order for the third spade to be drawn on the sixth draw, there must be exactly 2 spades in the first 5 draws and then a spade in the sixth draw. This means that in the first five card being drawn, there will be exactly 2 spades and 3 non-spades (cards from other suits).

Calculate the number of favorable outcomes

To calculate the number of these favorable outcomes, we will use combinations formula. In a combination, the order of items does not matter, making it suitable for this scenario since cards are drawn without replacement. First, calculate the number of combinations of choosing 2 spades out of 13 available spades: \(C(13,2) = \frac{13!}{(13-2)!2!} = \frac{13!}{11!2!} = 78\) Next, calculate the number of combinations of choosing 3 non-spades out of the remaining 39 cards (52 cards total minus the 13 spades): \(C(39,3) = \frac{39!}{(39-3)!3!} = \frac{39!}{36!3!}=9139\) Now, multiply these two results to obtain the number of possible favorable outcomes: \(78\times9139 = 712,542\)

Calculate the total number of possible outcomes

We need to calculate the total number of combinations of drawing 6 cards from a deck of 52 cards. This is given by: \(C(52,6) = \frac{52!}{(52-6)!6!}=\frac{52!}{46!6!} = 20,358,520\)

Divide the favorable outcomes by the total possible outcomes

Finally, divide the number of favorable outcomes by the total possible outcomes to get the probability: \(P = \frac{712,542}{20,358,520}\), which gives: \(P\approx0.03498\) The probability that the third spade will appear on the sixth draw is approximately 0.03498 or 3.498%.