Question

Let A1, . . . , An be a partition of the sample space Ω. Let X be a random variable. What is the formula for E(X) in terms of the conditional expected values E(X|A1), . . . , E(X|An)?

(b) Two coins are tossed. Then cards are drawn from a standard deck, with replacement, until the number of “face” cards drawn (a “face” card is a jack, queen, or king) equals the number of heads tossed. Let X = the number of cards drawn. Find E(X)

Answer #1

Probabilities with a deck of cards. There are 52 cards in a
standard deck of cards. There are 4 suits (Clubs, Hearts, Diamonds,
and Spades) and there are 13 cards in each suit. Clubs/Spades are
black, Hearts/Diamonds are red. There are 12 face cards. Face cards
are those with a Jack (J), King (K), or Queen (Q) on them. For this
question, we will consider the Ace (A) card to be a number card
(i.e., number 1). Then for each...

An experiment consists of selecting a card from a standard deck
of 52 cards, noting whether or not the card is an Face Card (Jack,
Queen or King) and returning the card to the deck. This experiment
is repeated 10 times. Let X be the random variable representing how
many times out of the 10 you observe a Face Card.
Suppose that you repeat the experiment 5070 times. Let Y be the
random variable representing the number of Face Cards...

A deck of cards has 52 cards with 4 suits (Hearts, Diamonds,
Spades, and Clubs) and 13 cards in each suit (Ace thru 10, Jack,
Queen, and King; the last three are considered face cards). A card
is drawn at random from a standard 52-card deck. What
is the probability that the card is a number card given the card is
black (Spades and Clubs)?
Group of answer choices
6/26
1 - 10/26
20/52
10/13

Sam is going to repeatedly select cards from a standard deck of
52 cards with replacement until he gets his 5th heart card.
Let Y be the number of cards he is drawing.
Let X1 be the number of spot cards (2, 3, 4, 5, 6, 7, 8, 9, 10) he
draws,
X2 be the number of face cards (Jack, Queen and King) he
draws,
and X3 be the number of aces he draws.
Find the JOINT PROBABILITY MASS FUNCTION...

Question 8: Consider a complete deck of 52 cards, not including
Jokers.
In such, an Ace=1, a King=13, a Queen=12 and a
Jack=11. All other cards are equal to face value. Assume that the
value of a card drawn from the deck is a random variable, X.
a. What is the probability distribution of X?
b. What is the mean of X?
c. What is the variance of X?
d. What is the standard deviation of X?

We draw one card at a time without replacement from the top of a
shuffled standard deck of cards and stop when we draw a Jack of any
suit. Let X be the number of cards we have drawn. What is P( X = 8
) ? Round to two decimals.

Let A1, A2, . . . , An be n independent events in a sample space
Ω, with respective probability pi = P (Ai). Give a simple
expression for the probability P(A1 ∪A2 ∪...∪An) in terms of p1,
p2, ..., pn. Let us now apply your result in a practical setting: a
robot undergoes n independent tests, which are such that for each
test the probability of failure is p. What is the probability that
the robot fails at least...

I toss 3 fair coins, and then re-toss all the ones that come up
tails. Let X denote the number of coins that come up heads on the
first toss, and let Y denote the number of re-tossed coins that
come up heads on the second toss. (Hence 0 ≤ X ≤ 3 and 0 ≤ Y ≤ 3 −
X.)
(a) Determine the joint pmf of X and Y , and use it to calculate
E(X + Y )....

A card is randomly selected from a standard, 52-card deck. The
denomination on the card is recorded so that the resulting sample
space is {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, K, Q, J}. (In other words,
we ignore the suits.) (a) (5 points) Given that the card selected
displays some number from 3 to 10 (inclusive), what is the
probability that the value on the card is not a multiple of 4? (b)
(5 points) Suppose...

Looking closely, you will see that the Jack of Spades and the
Jack of Hearts are revealing only one eye. These are the “one-eyed
Jacks” and are used as wild cards in a popular home-version of
poker, along with the four 2’s (or “Deuces”). The King
of Diamonds is also showing only one eye, so let us include it
among the wild cards. That gives us a total of 7 wild cards: the
four 2’s, the two one-eyed Jacks, and...

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