a card is drawn from a deck of 52 and recorded. then the card is replaced, and the deck is shuffled. this process is repeated 13 times,what is the probability that at least one of the cards drawn is a heart?
probability heart not drawn in each trial = 3/4
probability heart not drawn in 13 independent trials = (3/4)^13
probability that at least one heart is drawn = 1 - .75^13
By the way, independent because you always put the card back after each choice so 1/4 of the deck is always hearts
To find the probability that at least one of the cards drawn is a heart, we need to calculate the probability that none of the cards drawn are hearts and then subtract that from 1.
Let's break it down step by step:
Step 1: Probability of drawing a heart on the first draw
Since there are 13 hearts in a deck of 52 cards, the probability of drawing a heart on the first draw is 13/52.
Step 2: Probability of not drawing a heart on the first draw
Since we are replacing the card and shuffling the deck after each draw, the probability of not drawing a heart on the first draw is 1 - (13/52) = 39/52.
Step 3: Probability of not drawing a heart on any of the 13 draws
Since we are repeating this process 13 times, the probability of not drawing a heart on any of the 13 draws is (39/52)^13.
Step 4: Probability of at least one heart being drawn
To find the probability of at least one heart being drawn, we subtract the probability of not drawing a heart from 1:
1 - (39/52)^13 ≈ 1 - 0.0346 = 0.9654.
Therefore, the probability that at least one of the cards drawn is a heart is approximately 0.9654 or 96.54%.
To calculate the probability of at least one heart being drawn, we need to find the complement of the event that no hearts are drawn in any of the 13 draws.
The probability of drawing a card that is not a heart in a single draw is 39/52 because there are 39 cards that are not hearts out of the total 52 cards in the deck.
Now, let's find the probability of not drawing a single heart in any of the 13 draws:
(39/52) * (39/52) * (39/52) * ... * (39/52) (13 times) = (39/52)^13
To find the probability of at least one heart being drawn, we subtract this probability from 1:
1 - (39/52)^13
Now, let's calculate it:
1 - (39/52)^13 ≈ 0.9515
Therefore, the probability that at least one of the cards drawn is a heart is approximately 0.9515 or 95.15%.