Seven cards are selected at random from a 52-card deck. What is the probability that all of these cards are spades?

Question

Also what is the probability that at least one of these cards is a spade?

Answer 1

Hi John,

 

Please look at the answer I gave to the other problem you asked before reading this; it will make your understanding better.

 

In the standard deck, there are 13 spades, and 39 other cards out of 52. If you want to select 7 spades, you can calculate the ways to select 7 out of 13 spades and 0 of the rest of the deck, from a total of 7 cards from the whole deck.

 

P(7 spades) = ₁₃C₇ * ₃₉C₀ / ₅₂C₇

 

You could also find this by calculating each probability separately and multiplying:

13/52 * 12/51 * 11/50 * ... for all seven cards.

 

---

 

To find the probability of getting at least 1 spade, it's easier to look at the complement (opposite) of this. Rather than counting up the probability of 1 or 2 or 3 or ... or 7 spades, you can find the probability of getting less than 1 spade, or 0 spades, and subtracting from 100% (1).

 

You found the probability of getting no spades in the other problem you asked. Subtract that answer from 1 to find the probability of at least one.

 

P(X ≥ 1) = 1 - P(X = 0)