: Two cards are drawn from a well-shuffledordinary pack of 52 cards. The first card is replaced before selecting the second card. Find the following probabilities.

(i) Both are kings
(ii) First is king and second is queen
(iii) One is King and one is queen
(iv) Both are faced cards
(v) First is card of diamonds and the second is an ace.
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If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

Out of the 52 cards there are 4 kings, 4 Queens, 12 face cards, 13 diamonds and 4 aces.

1. 4/52 * 4/52 = ?
5. 13/52 * 4/52 = ?

Do the others with a similar process.

you are dealt two cards successively without replacement, from a deck of 52 playing cards. find the probability that both cards are black

To find the probabilities, we need to consider the total number of possible outcomes and the number of favorable outcomes for each case. Let's calculate the probabilities step by step:

Step 1: Calculate the total number of possible outcomes
When two cards are drawn and replaced, the total number of possible outcomes will be the product of the possible outcomes for each card drawn. Since there are 52 cards in a deck, the possible outcomes for each card drawn are also 52.

Total number of possible outcomes = 52 * 52 = 2,704

(i) Both are kings:
There are a total of 4 kings in a deck. Since the first card is replaced before drawing the second card, the favorable outcome of drawing a king remains the same for both cards.

Number of favorable outcomes = 4 * 4 = 16

Probability of drawing both kings = Number of favorable outcomes / Total number of possible outcomes
= 16 / 2,704

(ii) First is king and second is queen:
For the first card to be a king, there are 4 possible favorable outcomes. After replacing the first card, the remaining deck will have 52 cards, out of which 4 are queens.

Number of favorable outcomes = 4 * 4 = 16

Probability of drawing a king and a queen = Number of favorable outcomes / Total number of possible outcomes
= 16 / 2,704

(iii) One is king and one is queen:
To calculate this probability, we need to subtract the probabilities of the previous two cases from each other.

Probability of one king and one queen = Probability of (ii) - Probability of (i)

(iv) Both are faced cards:
Faced cards include the king, queen, and jack. Therefore, there are a total of 12 faced cards in a deck. The first card being replaced doesn't affect the favorable outcomes for the second card.

Number of favorable outcomes = 12 * 12 = 144

Probability of drawing both faced cards = Number of favorable outcomes / Total number of possible outcomes
= 144 / 2,704

(v) First is a card of diamonds and the second is an ace:
In a deck, there are 13 diamonds and 4 aces. Therefore, for the first card to be a diamond, there are 13 possible favorable outcomes. Since the first card is replaced, the favorable outcome for drawing an ace remains the same for the second card.

Number of favorable outcomes = 13 * 4 = 52

Probability of drawing a diamond and an ace = Number of favorable outcomes / Total number of possible outcomes
= 52 / 2,704

Now, you can calculate the probabilities by plugging the numbers into the formulas derived above.