Final answer:
If four cards are drawn from a standard deck of 52 cards and replaced, the probability of getting at least one heart is 175/256. If the cards are not replaced, the probability of getting at least one heart is 89/98.
Step-by-step explanation:
To calculate the probability of getting at least one heart, we need to consider two scenarios:
a) If the four cards are drawn with replacement:
The probability of getting a heart on any given draw is 1/4, since there are 13 hearts out of 52 cards in a deck.
The probability of not getting a heart on any given draw is 3/4, since there are 39 non-heart cards out of 52.
Since the draws are independent, we can multiply the probabilities together for four draws: (3/4) * (3/4) * (3/4) * (3/4) = 81/256.
The probability of getting at least one heart is 1 - (81/256) = 175/256.
b) If the four cards are drawn without replacement:
The probability of getting a heart on the first draw is 13/52.
The probability of getting a heart on the second draw is 12/51, since there are 12 hearts left out of the remaining 51 cards.
Similarly, the probability of getting a heart on the third draw is 11/50, and on the fourth draw is 10/49.
Again, we multiply these probabilities together: (13/52) * (12/51) * (11/50) * (10/49) = 9/98.
The probability of getting at least one heart is 1 - (9/98) = 89/98.