Seven cards are selected at random from a standard 52-card deck. What is the probability that none of these cards are spades?

Question

I really appreciate any help! Thank you!

Answer 1

Hi John,

 

In a standard deck of playing cards, there are 4 suits with 13 cards each. At the beginning of the drawing, there are 39 possible cards that are not spades, out of the total 52. Therefore, the probability of the first card not being a spade is

39/52 = 3/4

 

Having drawn one spade out of the deck, there are 12 left, out of 51 cards total, for a probability of

38/51

 

For the third card,

37/50

 

and so on, until you've drawn seven cards. To find the probability of all of them happening at once, multiply all the probabilities.

 

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You can also do this problem with a shortcut if you've worked with combinations before, where

_nC_x or (ⁿ_x) = n! / [(n-x)! x!]

 

If you're selecting 0 out of the 13 spades, and 7 out of the 39 other cards, you can do that

₁₃C₀ * ₃₉C₇ ways, out of ₅₂C₇ ways to select 7 cards from the whole deck. This gives you

 

₁₃C₀ * ₃₉C₇ / ₅₂C₇

 

which will give the same value as calculating the seven separate probabilities earlier.