Lacy draws a heart from standard deck of 52 cards. Without replacing the first c

Question

Lacy draws a heart from standard deck of 52 cards. Without replacing the first card, she then proceeds to draw a second card and gets a club. Are these events independent? Input Yes or No: Determine the probability of drawing a heart and then a club without replacement. Write your answer in decimal form, rounded to four decimal places as needed. Linda, draws a heart from a standard deck of 52 cards. She returns the heart to the deck, then draws a second card. Her second card is a club. Are these events Independent? Input Yes or No: Determine the probability of drawing a heart and then a club with replacement. Write your answer in decimal form, rounded to four decimal places as needed.

Explanation / Answer

Suppose,

A = getting the first card as a heart

B = getting the second card as a club

Then,

P(B|A) = 13/51

P( B | not A) = P(B | first card is a club). P(first card is club) + P(B | first card is not a club). P(first card is not a club)

= 12/51 * 13/52 + 13/51 * 39/52 = 12 /(51*4) + 39/(51*4) = 51/(5*4) = 1/4

So, these 2 events are NOT independent. Probability of B depends on A happens or not.

So, answer is NO for first part.

P( A and B) = P(first card as heart and second card as club) = P(second as club | first as heart). P(first as heart)

= 13/51 * 13/52 = 0.25 * 13/51 = 0.0637

For the case of "With replacment",

P(B | A) = P(second card is club | First card was heart) = 13/52 = P(B), since replacing the first card back to the deck completes the deck and hence picking the second card does not depend on first card outcome.

Hence, A and B are independent. Hence, answer is YES.

Also, P( A and B) = P(A) * P(B) = 13/52 * 13/52 = 1/16 = 0.0625

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