A card is pulled from a well shuffled deck of standard playing cards and then put back in the deck. A second draw is made from the deck. What is the probability that an ace was drawn both times?

4/52 * 4/52

the two draws are independent because the card was put back in

IF it were not put back
THEN it would have been 4/52 * 3/51 BUT THAT IS NOT WHAT THE QUESTION IS. In other words they are trying to trick you.

You draw two cards from a standard deck of playing cards without replacing the first one before drawing the second.

a. Are the two outcomes on the two cards independent? Why?
b. Find P(ace on 1st card and king on 2nd)
c. You repeat this problem but this time plan to put the first card back and reshuffle the deck before selecting the second card. Are the outcomes on the two cards independent?
d. Find P(ace on 1st card and king on 2nd

To find the probability of drawing an ace on both the first and second draws, we need to consider the probability of drawing an ace on the first draw, and then multiply it by the probability of drawing an ace on the second draw.

Step 1: Determine the probability of drawing an ace on the first draw.
A standard deck of playing cards contains four aces (one of each suit: hearts, diamonds, clubs, and spades) out of a total of 52 cards. So, the probability of drawing an ace on the first draw is 4/52, which simplifies to 1/13.

Step 2: Determine the probability of drawing an ace on the second draw.
Since the card from the first draw is put back into the deck, the deck remains unchanged for the second draw. Therefore, the probability of drawing an ace on the second draw is still 1/13, since there are still four aces in the deck and a total of 52 cards.

Step 3: Multiply the probabilities together.
To find the probability of both events occurring (drawing an ace on the first draw and the second draw), we multiply the individual probabilities:
P(Ace on both draws) = P(Ace on first draw) * P(Ace on second draw)
P(Ace on both draws) = (1/13) * (1/13)

Calculating this expression, we find that the probability of drawing an ace on both draws is:
P(Ace on both draws) = 1/169

Therefore, the probability of drawing an ace on both the first and second draws is 1/169 or approximately 0.0059 (rounded to four decimal places).