Two cards are drawn from a well-shuffled ordinary pack of 52 cards. The first card is replaced before selecting the second card. Find the following probabilities.
(iii) One is King and one is queen
(iv) Both are faced cards
a. Pr(K,Q)=4/52 * 4/52
b. Pr(F,F)=12/52*12/52 aces are not face
16/2704 =1/169 or 0.0059
144/2704 = 9/169 or 0.0533
To find the probabilities, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Given:
- Well-shuffled ordinary pack of 52 cards
- The first card is replaced before selecting the second card
(iii) One is King and one is Queen:
1. Determine the number of favorable outcomes:
Since there are 4 Kings and 4 Queens in a deck, we have 4 possible ways to choose a King and 4 possible ways to choose a Queen.
- The first card can be a King and the second card a Queen (4 Kings x 4 Queens = 16 favorable outcomes).
- The first card can be a Queen and the second card a King (4 Queens x 4 Kings = 16 favorable outcomes).
So, there are a total of 16 + 16 = 32 favorable outcomes.
2. Determine the total number of possible outcomes:
Since there are 52 cards in total and the first card is replaced before choosing the second card, each card is chosen independently.
- The first card can be any of the 52 cards.
- The second card can also be any of the 52 cards.
So, there are a total of 52 x 52 = 2,704 possible outcomes.
3. Calculate the probability:
The probability of having one King and one Queen is the ratio of favorable outcomes to total possible outcomes:
Probability = (number of favorable outcomes) / (total number of possible outcomes)
= 32 / 2,704
≈ 0.0118
Therefore, the probability of drawing one King and one Queen is approximately 0.0118, or 1.18%.
(iv) Both are faced cards:
1. Determine the number of favorable outcomes:
Faced cards include Kings, Queens, and Jacks. Since we need both drawn cards to be faced cards, the first card can be any of the 12 faced cards, and the second card can also be any of the 12 faced cards.
So, there are a total of 12 x 12 = 144 favorable outcomes.
2. Determine the total number of possible outcomes:
Since there are 52 cards in total and the first card is replaced before choosing the second card, each card is chosen independently.
- The first card can be any of the 52 cards.
- The second card can also be any of the 52 cards.
So, there are a total of 52 x 52 = 2,704 possible outcomes.
3. Calculate the probability:
The probability of drawing two faced cards is the ratio of favorable outcomes to total possible outcomes:
Probability = (number of favorable outcomes) / (total number of possible outcomes)
= 144 / 2,704
≈ 0.0532
Therefore, the probability of drawing two faced cards is approximately 0.0532, or 5.32%.