A card is drawn from an ordinary deck of 52 cards, and the result is recorded on paper. The card is then returned to the deck and another card is drawn and recorded. Find the probability that the following occurs:

a) the first card is a diamond
b) the second card is a diamond
c) both cards are diamonds
d) neither card is a diamond

To find the probability in each case, we need to consider the total number of possible outcomes and the number of favorable outcomes.

a) The probability that the first card is a diamond:
There are 13 diamonds in a deck of 52 cards. Therefore, the probability of drawing a diamond as the first card is:

Number of favorable outcomes = 13 diamonds
Total number of possible outcomes = 52 cards

Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 13 / 52
Probability = 1/4 or 0.25

b) The probability that the second card is a diamond:
Since the first card is returned to the deck, we still have 13 diamonds in the deck. The probability of drawing a diamond as the second card is the same as for the first card:

Probability = 1/4 or 0.25

c) The probability that both cards are diamonds:
To find the probability of both cards being diamonds, we multiply the probabilities of each individual event (drawing a diamond card):

Probability of the first card being a diamond = 1/4
Probability of the second card being a diamond = 1/4

Probability of both cards being diamonds = (1/4) * (1/4)
Probability = 1/16 or 0.0625

d) The probability that neither card is a diamond:
To find the probability of neither card being a diamond, we subtract the probability of getting a diamond from 1.

Probability of neither card being a diamond = 1 - Probability of the first card being a diamond
Probability of neither card being a diamond = 1 - 1/4
Probability of neither card being a diamond = 3/4 or 0.75

To find the probability of each event, we need to use the total number of cards in the deck, as well as the number of favorable outcomes for each event.

a) The probability that the first card is a diamond:
- Total number of cards in the deck = 52
- Number of diamonds in the deck = 13 (since there are 13 diamonds in a regular deck)
Therefore, the probability of the first card being a diamond is 13/52, which simplifies to 1/4.

b) The probability that the second card is a diamond:
Since the first card is returned to the deck, the total number of cards remains 52, but the number of diamonds is reduced to 12 (as one diamond has already been drawn).
Therefore, the probability of the second card being a diamond is 12/52, which simplifies to 3/13.

c) The probability that both cards are diamonds:
The probability of drawing a diamond on the first card is 1/4, and since the card is returned to the deck, the probability of drawing a diamond on the second card is also 1/4.
Therefore, the probability that both cards are diamonds is (1/4) * (1/4) = 1/16.

d) The probability that neither card is a diamond:
The probability of the first card not being a diamond is 1 - 1/4 = 3/4.
Since the first card is returned to the deck, the probability of the second card not being a diamond is also 3/4.
Therefore, the probability that neither card is a diamond is (3/4) * (3/4) = 9/16.

a) 13/52 = ?

b) 13/52 = ?

c) If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

d) one card not a diamond = (52-13)/52. Take it from here.