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.

Answer 1

Without replacing the first card, the events are dependent with a combined probability of 0.0588. With the replacement of the first card before drawing the second, the events are independent with a probability of 0.0625.

Step-by-step explanation:

The probability of drawing a heart and then a club without replacement from a standard deck of 52 cards is dependent on the outcome of the first draw. Since we do not replace the first card (a heart), there are now 51 cards remaining in the deck with 12 of those being clubs. The probability of drawing a heart first is 13/52, and the subsequent chance of drawing a club is 12/51. The combined probability of these dependent events is therefore (13/52) * (12/51) = 0.0588, rounded to four decimal places.

Conversely, when Lacy replaces the first card before drawing the second, the two events are independent. The probability of drawing a heart remains 13/52, and the probability of drawing a club also remains 13/52, as the deck goes back to its original composition of 52 cards. The combined probability of drawing a heart followed by a club with replacement is therefore (13/52) * (13/52) = 0.0625.

Answering the question:

• Are these events independent without replacement? No
• Are these events independent with replacement? Yes