Question

three cards are drawn in succession from a deck without replacement. find the probability distribution for the number of hearts

Answer 1

There are 13 heart cards in a deck of cards and there are 39 cards which are not heart.

To find the probability distribution of selecting a card of heart.

Let x denotes the number of heart cards in a draw of 3 cards without replacement, so x could take values as 0, 1, 2, and 3.

Probability of selection a heart card $= \frac{13}{52}$

Probability of selection two heart cards in succession (without replacement) $= \frac{13}{52} \times \frac{12}{51}$

Probability of selection three heart cards in succession (without replacement) $= \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50}$

Probability of selection a non-heart card $= \frac{39}{52}$

Probability of selection two non-heart cards in succession (without replacement) $= \frac{39}{52} \times \frac{38}{51}$

Probability of selection three non-heart cards in succession (without replacement) $= \frac{39}{52} \times \frac{38}{51} \times \frac{37}{50}$

Hence P(x=0) =selecting 3 heart cards in succession (without replacement) $= \frac{39}{52} \times \frac{38}{51} \times \frac{37}{50} = 0.4135$

P(x=1) = selecting 1 heart card and 2 non-heart cards in succession (without replacement) $= \frac{13}{52} \times \frac{39}{51} \times \frac{38}{50} = 0.1452$

P(x=2)=selecting 2 heart card and 1 non-heart cards in succession (without replacement) $= \frac{13}{52} \times \frac{12}{51} \times \frac{39}{50} = 0.0458$

P(x=3) = selecting 3 heart cards in succession (without replacement) $= \frac{13}{52} \times \frac{12}{51} \times \frac{11}{50} = 0.0129$

Probability distribution table: