Question

two cards are drawn successively without replacement from a shuffled deck of cards. make a probability distribution table where random variable x represents the number of heart

Answer 1

Answer on Question #70398, Math / Statistics and Probability

Two cards are drawn successively without replacement from a shuffled deck of cards. Make a probability distribution table where random variable $x$ represents the number of heart.

Solution

There are 13 hearts $\blacktriangledown$ in a full deck of 52 cards.

Random variable $x$ represents the number of heart.

WITHOUT REPLACEMENT: If you draw two cards from the deck without replacement, what is the probability that there will be no hearts?

$$P(X = 0) = P(\text{no heart and no heart}) = \left(\frac{39}{52}\right) \left(\frac{38}{51}\right) = \frac{19}{34}$$

WITHOUT REPLACEMENT: If you draw two cards from the deck without replacement, what is the probability that there will be the only heart?

$$\begin{array}{l} P(X = 1) = P(\text{heart and no heart}) + P(\text{no heart and heart}) = \\ = \left(\frac{39}{52}\right) \left(\frac{13}{51}\right) + \left(\frac{13}{52}\right) \left(\frac{39}{51}\right) = \frac{13}{34} \end{array}$$

WITHOUT REPLACEMENT: If you draw two cards from the deck without replacement, what is the probability that they will both be hearts?

$$P(X = 2) = P(\text{heart and heart}) = \left(\frac{13}{52}\right) \left(\frac{12}{51}\right) = \frac{1}{17}$$

Check

$$P(X = 0) + P(X = 1) + P(X = 2) = \frac{19}{34} + \frac{13}{34} + \frac{1}{17} = 1$$

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