Question

A standard deck consists of 52 cards of which 4 are aces, 4 are kings, and 12 (including the four kings) are "face cards" (Jacks, Queens, and Kings). Cards are dealt at random without replacement from a standard deck till all the cards have been dealt. Find the expectation of the following. Each can be done with almost no calculation if you use symmetry.

a) The number of aces among the first 5 cards

b) The number of face cards that do not appear among the first 13 cards

c) The number of aces among the first 5 cards minus the number of kings among the last 5 cards

d) The number of cards before the first ace

e) The number of cards strictly in between the first ace and the last ace

f) The number of face cards before the first ace

Answer 1

(a)

Let, X = The number of aces among the first 5 cards

Then,

(b)

Let,

X=Number of face cards that do not appear among first 13 cards

(c)

Let,

X = Number of aces among first 5 cards, and

Y = Number of kings among first 5 cards

Total number of aces among all 52 cards = Total number of kings among all 52 cards = 4

Therefore,

Hense,

This is easily understandable as X-Y is symmetric with mean 0.

(d)

Let,

X=Number of cards before the first ace