a standard deck of playing cards contaons 52 cards in for suits of 13 cards each. two suits are red and two suits are black. find each probability. assume the first card is replaced before the second is drawn.
(balck,Queen) (black,Diamond) (jack,Queen)
first one:
prob(black) = 1/2
prob(queen) = 4/52 = 1/13
prob(black queen in that order) = (1/2)(1/13) = 1/26
There are several interpretations possible for this question.
e.g.
- what if you first card is the black queen? can the second card be anything?
- does the order matter, that is, do you consider (black, queen) the same as (queen, black) ?
etc.
To find the probability of each event, we need to determine the number of favorable outcomes (the cards that meet the given condition) and the total number of possible outcomes (the total number of cards).
1. Probability of drawing a black Queen:
- There are 26 black cards in total since two suits are black (Clubs and Spades), and each suit contains 13 cards.
- There are 4 Queens in the deck, and only one of them is black.
- Therefore, the number of favorable outcomes is 1 (the black Queen) and the total number of possible outcomes is 52 (the total number of cards).
- So, the probability of drawing a black Queen is 1/52.
2. Probability of drawing a black Diamond:
- There are 26 black cards in total since two suits are black (Clubs and Spades), and each suit contains 13 cards.
- There are 1 black Diamond in the deck.
- Therefore, the number of favorable outcomes is 1 (the black Diamond) and the total number of possible outcomes is 52 (the total number of cards).
- So, the probability of drawing a black Diamond is 1/52.
3. Probability of drawing a Jack and a Queen:
- There are 4 Jacks in the deck and 4 Queens.
- Since we are replacing the first card before drawing the second, the probabilities are independent, and we multiply the individual probabilities.
- The probability of drawing a Jack is 4/52 (4 Jacks in 52 cards).
- The probability of drawing a Queen is also 4/52 (4 Queens in 52 cards).
- Therefore, the probability of drawing a Jack and then a Queen is (4/52) * (4/52) = 16/2704.
So, the probabilities are:
1. Probability of drawing a black Queen: 1/52.
2. Probability of drawing a black Diamond: 1/52.
3. Probability of drawing a Jack and then a Queen: 16/2704.
To find each probability, we need to calculate the number of favorable outcomes (desired outcomes) and divide it by the number of possible outcomes.
Probability of drawing a black Queen:
In a standard deck, there are 4 Queens, and 2 of them are black (Clubs and Spades). Therefore, the number of favorable outcomes is 2. Since we are assuming the first card is replaced before the second is drawn, the total number of possible outcomes is also 52. So the probability of drawing a black Queen is:
Probability = Favorable outcomes / Total outcomes = 2/52 = 1/26
Probability of drawing a black Diamond:
In a standard deck, there are 13 Diamonds, and 1 of them is black (since the other 3 suits are red - Hearts, Clubs, and Spades). Therefore, the number of favorable outcomes is 1. Again, the total number of possible outcomes is 52. So the probability of drawing a black Diamond is:
Probability = Favorable outcomes / Total outcomes = 1/52
Probability of drawing a Jack and a Queen:
In a standard deck, there are 4 Jacks and 4 Queens. Since we are assuming the first card is replaced before the second is drawn, we have 4 choices for the first card and 4 choices for the second card. Therefore, the number of favorable outcomes for drawing a Jack and a Queen is 4 x 4 = 16. The total number of possible outcomes remains 52 x 52. So the probability of drawing a Jack and a Queen is:
Probability = Favorable outcomes / Total outcomes = 16/(52 x 52)
Simplifying the expression further is not possible without knowing the context or any specific event.
So in summary:
- Probability of drawing a black Queen = 1/26
- Probability of drawing a black Diamond = 1/52
- Probability of drawing a Jack and a Queen = 16/(52 x 52)