You draw two cards from a standard deck without replacement and without looking at them. Then...

You draw two cards from a standard deck without replacement and without looking at them. Then the third card is drawn and it is revealed: it is the ace of hearts. What is the conditional probability that your two cards, formerly hidden, are aces?

Solutions

Expert Solution

There are 4 aces and 48 other cards in a deck.

There are three cases possible :

First two card are aces or one of first two cards is ace or none of first two card is ace.

Case: First two card are aces

The probabilty of getting third ace and first two aces is

P(first two aces and third ace) =(4/52) * (3/51) * (2/50) = 24 /(52*51*50)

Case: One of first two cards is ace

The probabilty of getting third ace is

P(one of first two is an ace and third ace) =2 * (4/52) * (48/51) * (3/50) = 1152 / (52*51*50)

Case: None of first two cards is ace

The probabilty of getting third ace is

P(none of first two is an ace and third ace) = (48/52) * (47/51) * (3/50) = 6768 / (52*51*50)

The probability that third card is an ace is:

P(third ace) = [24 /(52*51*50)]+ [1152 / (52*51*50)] + [9024 / (52*51*50)] = 10200 / (52*51*50)

The conditional probability that your two cards, formerly hidden, are aces is

P(first two cards ace |third ace) = P(first two aces and third ace) / P(third ace) = 24 / 10200 = 0.0024

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Suppose that we draw two cards from a standard deck without replacement. What is the probability...

Suppose that we draw two cards from a standard deck without replacement. What is the probability that we draw the king of hearts and then any other heart? White your answer in decimal form, rounded to three decimal places.

draw 20 cards without replacement from a shuffled, standard deck of 52 cards. What is the...

draw 20 cards without replacement from a shuffled, standard deck of 52 cards. What is the conditional probability P (12th card and 20th card is a heart 3rd card is a 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...

Two cards will be dealt, one at a time and without replacement, from a standard deck...

Two cards will be dealt, one at a time and without replacement, from a standard deck of 52 playing cards. In a standard deck, there are 4 Jacks. Answer the following: a) What is the probability the second card dealt will be a Jack? P(second card will be Jack) = _______ b) What is the probability both cards dealt will be Jacks? P(both dealt cards will be Jacks) = _______ c) What is the probability neither card dealt will be...

2 cards are drawn without replacement from a standard deck. draw a venn diagram that relates...

2 cards are drawn without replacement from a standard deck. draw a venn diagram that relates to the following: 1. Both cards are from the same suit 2. Both cards have the same rank 3. One card is a face card and the other card is a number card 4. Both cards are from a red suit

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

Question 2. Draw three cards without replacement from a deck of cards and let X be...

Question 2. Draw three cards without replacement from a deck of cards and let X be the number of spades drawn. Sketch the pmf of X and compute E(X). Question 3. A fair coin is flipped n times. What is the probability of getting a total of k heads if a) The first flip shows heads b) The first flip shows tails c) At least one flip shows heads

Suppose we draw 5 cards at random, without replacement, from a deck of 52 cards (such...

Suppose we draw 5 cards at random, without replacement, from a deck of 52 cards (such a deck includes 4 Queens). Let X denote the number of Queens drawn. Define some indicator random variables X1,..., X5 so that X = X1+:::+X5. Then use the random variables you created to find E(X). Also calculate the E[X] if this time 5 cards are drawn with replacement.

You draw two cards from a standard deck of 52 cards, but before you draw the...

You draw two cards from a standard deck of 52 cards, but before you draw the second card, you put the first one back and reshuffle the deck. (Round your answer to three decimal places) 1) Find P(Ace on first card and Red card on second card) 2) Find P(Ace and King in either order) 3) If you do not replace the first card before drawing the second card, Find P(Ace on first card and King on second card)

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?

Subjects