Problems tagged with "conditional independence"

Problem #46

Tags: conditional independence

Recall that a deck of 52 cards contains:

Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Also recall that Hearts and Diamonds are red, while Clubs and Spades are black.

Part 1)

Suppose a single card is drawn at random.

Let \(A\) be the event that the card is a heart. Let \(B\) be the event that the card is a 5.

Are \(A\) and \(B\) independent events?

Solution

Yes, they are independent.

Part 2)

Suppose two cards are drawn at random (without replacing them into the deck).

Let \(A\) be the event that the second card is a heart. Let \(B\) be the event that the first card is red.

Are \(A\) and \(B\) independent events?

Solution

No, they are not.

Part 3)

Suppose two cards are drawn at random (without replacing them into the deck).

Let \(A\) be the event that the second card is a heart. Let \(B\) be the event that the second card is a diamond. Let \(C\) be the event that the first card is face card.

Are \(A\) and \(B\) conditionally independent given \(C\)?

Solution

No, they are not.

Problem #58

Tags: conditional independence

Suppose that a deck of cards has some cards missing. Namely, both the Ace of Spades and Ace of Clubs are missing, leaving 50 cards remaining.

Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A

Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K

Also recall that Hearts and Diamonds are red, while Clubs and Spades are black.

Part 1)

Suppose a single card is drawn at random.

Let \(A\) be the event that the card is a heart. Let \(B\) be the event that the card is an Ace.

Are \(A\) and \(B\) independent events?

Solution

No, they are not.

Part 2)

Suppose a single card is drawn at random.

Let \(A\) be the event that the card is a red. Let \(B\) be the event that the card is a heart. Let \(C\) be the event that the card is an ace.

Are \(A\) and \(B\) conditionally independent given \(C\)?

Solution

Yes, they are.

Part 3)

Suppose a single card is drawn at random.

Let \(A\) be the event that the card is a King. Let \(B\) be the event that the card is red. Let \(C\) be the event that the card is not a numbered card (that is, it is a J, Q, K, or A).

Are \(A\) and \(B\) conditionally independent given \(C\)?

Solution

No, they are not conditionally independent.