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If you are dealt 4 cards from a shuffled deck of 52 cards,find the probability that all 4 cards are diamonds The probability is Round to six decimal places as needed.)

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Answer:

The probability that all 4 cards are diamonds is 0.002641.

Step-by-step explanation:

We know that:

  • There are 52 cards in total
  • There are 13 diamond cards
  • We want to calculate the probability of a compound event without replacement

Denoting P(DDDD) as the probability that all 4 cards are diamonds, we can use the following formula:

[tex]P(DDDD)= \frac{D}{T}*\frac{D-1}{T-1}*\frac{D-2}{T-2}*\frac{D-3}{T-3}[/tex]

Where:

D is the total number of diamond cards in the deck, and

T is the total number of cards in the deck.

What we're doing is multiplying the probability that the first card is diamond by the probability that the second card is diamond, and so on. But as we're working without replacement the probability of each successive card has to take into account that the deck has one card less than before.

Replacing in the formula with the actual values we have:

[tex]P(DDDD)= \frac{13}{52}*\frac{12}{51}*\frac{11}{50}*\frac{10}{49}[/tex]

[tex]P(DDDD)=\frac{11}{4165}=0.002641[/tex]

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