Four cards are drawn from a standard deck. No cars are returned.
A) which is more likely:
Event A: drawing a straight( four consecutive cards e.g. 5,6,7,8)?
Event B: drawing four consecutive Queens?
B) Explain how your answer changed it cards are returned after each draw
A) Given that four cards are drawn from a standard deck without replacement, the probability of drawing a straight is higher than the probability of drawing four consecutive Queens.
To calculate the probability of drawing a straight, we can consider the number of ways a straight can occur and divide it by the total number of possible outcomes. There are 10 possible straights in a standard deck (A,2,3,4; 2,3,4,5; ...; 10,J,Q,K), and the total number of possible outcomes without replacement is the number of ways to choose 4 cards out of 52, which is given by the combination formula: C(52, 4) = 52! / (4! * (52-4)!). Therefore, the probability of drawing a straight is: P(A) = 10 / C(52, 4).
On the other hand, the probability of drawing four consecutive Queens is given by the number of ways four Queens can be drawn divided by the total number of possible outcomes without replacement. There is only one way to draw four consecutive Queens, and the total number of possible outcomes without replacement is C(52, 4). Therefore, the probability of drawing four consecutive Queens is: P(B) = 1 / C(52, 4).
Since P(A) > P(B), it is more likely to draw a straight than to draw four consecutive Queens.
B) If cards are returned after each draw, the probabilities of drawing a straight and drawing four consecutive Queens would change. Returning cards after each draw means that the deck is effectively reset to its original composition before each draw, so the probability of drawing any particular card remains the same at each draw.
Drawing a straight: Each draw is independent of the previous draws because the cards are returned, maintaining a uniform distribution in the deck. Therefore, the probability of drawing a straight is the product of the probabilities of drawing each card of the straight consecutively. For example, the probability of drawing 5,6,7,8 in order would be (4/52) * (4/52) * (4/52) * (4/52).
Drawing four consecutive Queens: With cards returned after each draw, the probability of drawing four consecutive Queens would remain the same as drawing any four specific cards in order: (4/52) * (4/52) * (4/52) * (4/52).
If cards are returned after each draw, the probabilities of drawing a straight and drawing four consecutive Queens would remain the same as calculating the probabilities for drawing any specific set of four cards in order, which is much lower than drawing a straight in general.
A) If no cards are returned after each draw, let's calculate the probabilities for Event A and Event B:
Event A: Drawing a straight (four consecutive cards)
To calculate the probability, we need to consider the number of possible combinations that result in a straight and divide it by the total number of possible combinations.
There are 10 possible combinations for a straight in a standard deck:
1. A, 2, 3, 4
2. 2, 3, 4, 5
3. 3, 4, 5, 6
4. 4, 5, 6, 7
5. 5, 6, 7, 8
6. 6, 7, 8, 9
7. 7, 8, 9, 10
8. 8, 9, 10, J
9. 9, 10, J, Q
10. 10, J, Q, K
The total number of possible combinations when drawing four cards from a deck is given by choosing 4 cards out of a total of 52 cards, which is calculated using binomial coefficient notation as C(52, 4) = 52! / (4! * (52 - 4)!).
Therefore, the probability of drawing a straight is 10 / C(52, 4).
Event B: Drawing four consecutive Queens
To calculate the probability of this event, we need to consider how many combinations result in drawing four consecutive Queens and divide it by the total number of possible combinations.
Since there are only four Queens in a standard deck, the only way to obtain four consecutive Queens would be to draw them in sequence. Therefore, there is only 1 possible combination for Event B.
The total number of possible combinations when drawing four cards from a deck remains the same as above, C(52, 4) = 52! / (4! * (52 - 4)!).
Therefore, the probability of drawing four consecutive Queens is 1 / C(52, 4).
Comparing the probabilities:
If you calculate the probabilities, you will find that the probability of Event A (drawing a straight) is higher than the probability of Event B (drawing four consecutive Queens).
B) If the cards are returned after each draw, the probabilities for Event A and Event B would change:
For Event A (drawing a straight), the probability would remain the same since each draw is independent and the probability of drawing a straight is dependent on the number of possible combinations.
For Event B (drawing four consecutive Queens), if the cards are returned after each draw, then the probability would increase. In the initial scenario, there was only 1 combination for Event B, but with cards returned, there would be multiple ways to obtain four consecutive Queens. Each time a Queen is drawn, it would be returned to the deck, allowing for the possibility of drawing another Queen in the next draw.
Therefore, if the cards are returned after each draw, the probability of Event B (drawing four consecutive Queens) would be higher compared to the scenario where cards are not returned.