A standard deck of playing cards has 52 cards in four suits (two red, two black)

Question

A standard deck of playing cards has 52 cards in four suits (two red, two
black), each suit with cards numbered 2{10 and ace cards" labelled jack,
queen, king, and ace. A pinochle deck" has 48 cards: each of the four suits
has only 9 and 10 plus the face cards, and each card appears twice in the
deck.

Assuming in each case that the relevant deck is well-shued, identify the fol-
lowing probabilities. Show your work by making it clear where each number
in your probability comes from.

a. p(R=ace j D=standard): In a standard deck, the probability of draw-
ing any card whose rank is `ace'

b. p(R=ace j D=pinochle): In a pinochle deck, the probability of drawing
any card whose rank is `ace'

c. p(F=true, C=red j D=pinochle): the probability of drawing a card
whose rank is one of the face cards and whose suit is one of the red
suits, from a pinochle deck

d. p(R=A, S=H j F=true): the probability of drawing the ace of hearts
given that the card drawn is a face card

e. If there are 3 standard decks and 1 pinochle deck in a pile, and you
pick one at random before drawing a card from that deck, what is
p(F=true), i.e. the overall probability of drawing a face card?

Explanation / Answer

Probalility = Favorable / Total

1) Total = 52

Fav = Total ace cards = 1 + 1 + 1 + 1 = 4

P = 4/52

2)

Total = 48

Fav = Total ace cards = 2 + 2 + 2 + 2 = 8

P = 8/42

3)

Total = 52

Fav = Total face cards with red color = 4 + 4 = 8

P = 8/52

4) Not given which deck, therefore, solving for both

Note its given that face card is there, therefore sample space or total is reduced

STANDARD DeCK

Total = 4 + 4 + 4 + 4 = 16

Fav = Ace of Hearts = 1

P = 1/16

pinochle DeCK

Total = 8 + 8 + 8 + 8 = 32

Fav = Ace of Hearts = 1

P = 1/32

5) p = p(choose standard) * p(face when standard) + p(choose pinochle) * p(face when pinochle)

p(choose standard) = 3/4

p(choose pinochle) = 1/4

p(face when standard)

total face = 16

total = 52

p(face when standard) = 16/52

p(face when pinochle)

total face = 32

total = 48

p(face when pinochle) = 32/48

please comment for any further assistance

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