Suppose you shuffle a deck of cards and set it down on the table. (a) What...

Suppose you shuffle a deck of cards and set it down on the table. (a) What is the probability that the top ten cards contain no aces? (There are four aces in the whole deck). (b) What is the probability that the top four cards are all the same suit? (There are four suits, and thirteen cards belonging to each suit). (c) What is the probability that the top four cards contain (exactly one) pair of the same card? (There are thirteen different cards and each occurs four times).

Expert Solution

Given, there are a deck of cards, 52 cards- 4 different suits, each suit containing 13 different cards

ANSWER a:

Probability that the top ten cards contain NO ACES = ( 48C10 / 52C10 )

= ( 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 x 40 x 39 ) / ( 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 )

= 246/595

( SINCE, 48C10 - In 52 cards, 4 Aces are there. So, 52-4 = 48 )

ANSWER b:

The required probability = ( 4 x 13C4 ) / 52C4 = ( 4 x 10 x 11 x 12 x 13) / ( 52 x 51 x 50 x 49 ) = 44/4165

( Since, there are 4 suits, that one suit could be any of the 4.

13C4 - the 4 cards of any given suit can be chosen in 13C4 ways)

ANSWER c:

The required probability= ( 13 x 4C2 x 48C2 ) / 52C4 = ( 13 x 6 x 48 x 47 x 4! ) / ( 2 x 49 x 50 x 51 x 52 ) = 6768/20825

( SINCE, there are 13 different numbers

the 2 numbers can be from any of the 4 suits, i.e; 4C2

the remaining 2 numbers can be any 2 of the (52-4)=48 cards, 48C2)

orchestra answered 2 years ago

A deck of 11 cards contains 4 red and 7 green cards. Shuffle this deck, then...

A deck of 11 cards contains 4 red and 7 green cards. Shuffle this deck, then deal 5 cards to Alice and 6 cards to Bob (without any replacement, of course). Let X be the number of red cards that Alice gets, and Y the number of red cards that Bob gets. (a) Write down the joint probability mass function of X and Y . (Write a formula rather than a table.) (b) Are X and Y independent? (c) Compute...

Find s shuffle of a deck of 13 cards that requires 42 repeats to return to...

Find s shuffle of a deck of 13 cards that requires 42 repeats to return to the original order. How about one requiring 30 repeats?

Cards: Suppose you draw one card from a single deck of cards. (a) What is the...

Cards: Suppose you draw one card from a single deck of cards. (a) What is the probability that you draw an queen? Round your answer to 3 significant digits*. (b) What is the probability that you draw a heart? Round your answer to 3 significant digits*. (c) What is the probability that you draw the queen of hearts? Round your answer to 3 significant digits*. ............................................... *Significant Digits: Here are some probabilities expressed to 3 significant digits. You start counting...

Imagine I offer you the following gamble. I shuffle an ordinary deck of cards (52 cards of 4 suits; no “jokers”). I draw one card from the deck

Imagine I offer you the following gamble. I shuffle an ordinary deck of cards (52 cards of 4 suits; no “jokers”). I draw one card from the deck. If I draw a Heart, you win $8. If I draw any other suit, you lose (pay me) $2 A. What’s the probability that you’d win the bet (i.e., I draw a Heart)? What’s the probability that you’d lose the bet? B.What’s the expected value of the gamble? (For this problem,...

Suppose you have a deck of 40 cards, containing 10 red cards and 30 black cards....

Suppose you have a deck of 40 cards, containing 10 red cards and 30 black cards. A player draws 7 cards from the deck and places them into their hand. The remaining deck of 33 cards has 5 red cards and 5 black cards in the top 10 cards. The other 23 cards in the remaining deck are unknown color. What is the probability of the player's hand having 0,1,2,3,4,5 red cards?

Now suppose that you shuffle the deck thoroughly and deal a standard five-card hand without replacement....

Now suppose that you shuffle the deck thoroughly and deal a standard five-card hand without replacement. Give the probability that the hand: (a) has only face cards (Jack, Queen, King); (b) has only red cards and does not contain an Ace; (c) is a Flush (all of one suit) containing only non-face cards; (d) contains neither an even-numbered card (2,4,6,8,10) nor a Spade. (e) contains at least one Ace and at least one Club.

Suppose three cards are drawn without replacement from a standard deck of cards. A standard deck...

Suppose three cards are drawn without replacement from a standard deck of cards. A standard deck of cards contains 52 cards, each card is one of 4 types (hearts, diamonds, spades, clubs) and cards within each type are labelled 2, 3, 4, …, 10, J, Q, K, A. Compute the probability for each of the following. a. All three cards selected is a Heart. b. All three cards selected is a King. c. None of the three cards is either...

You take 2 cards - without replacement - from a deck of cards. What is the...

You take 2 cards - without replacement - from a deck of cards. What is the probability of at least one ace and at least one spade?

Suppose that you select 2 cards without replacement from a standard deck of 52 playing cards....

Suppose that you select 2 cards without replacement from a standard deck of 52 playing cards. a) If the first card that you select is NOT a heart, what is the probability that the second card that you select IS a heart? b) If the first card that you select IS a six, what is the probability that the second card that you select IS a diamond? PLEASE SHOW ALL WORK

Suppose you are dealt 5 random cards from a standard deck of 52 cards, where all...

Suppose you are dealt 5 random cards from a standard deck of 52 cards, where all cards are equally likely to appear. (a) What is your outcome space? (b) What is the probability that you receive the ace of hearts? (c) Let AH be the event that you receive the ace of hearts, AC the event that you receive the ace of clubs, AD the event that you receive the ace of diamonds, and AS the event that you receive...

Question