A card is drawn at random from a standard deck of 52 cards. Find the following conditional probabilities. a) The card is a diamond, given that it is red. b) The card is red, given that it is a diamond. c) The card is a nine, given that it is red. d) The card is a king, given that it is a face card.

Question

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Respuesta :

berno berno · Answer 1 · 2020-04-27T17:20:04+02:00

Answer:

[tex]a.\frac{1}{13}\\b.\frac{1}{2}\\c.\frac{1}{13}\\d.\frac{1}{13}[/tex]

Step-by-step explanation:

P(A|B) = P(A and B) / P(B)

Total number of cards = 52

a)

Let A denotes the event that the card is a diamond and B denotes the card is red.

[tex]P\left ( A|B \right )=\frac{P\left ( A\cap B \right )}{P(B)}\\=\frac{\frac{2}{52}}{\frac{26}{52}}\\=\frac{1}{13}[/tex]

b)

Let A denotes the event that the card is red and B denotes the card is diamond.

[tex]P\left ( A|B \right )=\frac{P\left ( A\cap B \right )}{P(B)}=\frac{\frac{2}{52}}{\frac{4}{52}}\\=\frac{1}{2}[/tex]

c)

Let A denotes the event that the card is nine and B denotes the card is red.

[tex]P\left ( A|B \right )=\frac{P\left ( A\cap B \right )}{P(B)}=\frac{\frac{2}{52}}{\frac{26}{52}}\\=\frac{1}{13}[/tex]

d)

Let A denotes the event that the card is a king and B denotes the card is a face card.

[tex]P\left ( A|B \right )=\frac{P\left ( A\cap B \right )}{P(B)}=\frac{\frac{4}{52}}{\frac{12}{52}}\\=\frac{1}{13}[/tex]