suppose that 2 cards are randomly selected from a standard 52 card deck.
A) whats the probability the first card is a club and the second is a club if the sampling is done without replacement?
B) if both cards are clubs but it is done with replacement?
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
A) 13/52 * 12/51 = ?
B) 13/52 * 13/52 = ?
A) To calculate the probability that the first card is a club and the second card is a club without replacement, we first determine the number of favorable outcomes and then divide by the total number of possible outcomes.
Step 1: Determine the number of favorable outcomes.
There are 13 clubs in a standard deck of 52 cards. The first card being a club leaves 12 clubs in the remaining 51 cards, so there are 12 favorable outcomes.
Step 2: Determine the total number of possible outcomes.
When the first card is drawn, there are 52 cards in the deck. After the first card, there are 51 cards remaining.
Step 3: Calculate the probability.
The probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
P(First card is a club AND Second card is a club) = (12/52) * (11/51) = 0.0588 or 5.88%
Therefore, the probability that the first card is a club and the second card is a club, when the sampling is done without replacement, is approximately 0.0588 or 5.88%.
B) If both cards are clubs and the sampling is done with replacement, the probability remains the same for each draw because each draw is independent of the previous draw.
Step 1: Determine the probability of drawing a club.
There are 13 clubs in a standard deck of 52 cards, so the probability of drawing a club in each of the two draws is 13/52.
Step 2: Calculate the probability of both cards being clubs.
Since the draws are independent, we multiply the probabilities of each draw.
P(Both cards are clubs) = (13/52) * (13/52) = 169/2704 = 0.0625 or 6.25%
Therefore, the probability that both cards are clubs when the sampling is done with replacement is approximately 0.0625 or 6.25%.
To solve both parts of the question, we need to determine the total number of possible outcomes and the number of favorable outcomes for each scenario.
A) Without replacement:
When the first card is chosen, there are 52 cards in the deck. Since we want the first card to be a club, there are 13 clubs available. After the first card is selected, there remains a total of 51 cards, with 12 remaining clubs. Therefore, the probability of the second card being a club after the first card is a club is 12/51.
B) With replacement:
In this scenario, after the first card is chosen, it is returned to the deck before the second card is drawn. This means that every time a card is chosen, there are still 52 cards in the deck, and the number of clubs remains at 13. Therefore, the probability of the second card being a club is still 13/52.
To summarize:
A) Without replacement: P(The first card is a club and the second card is a club) = 13/52 * 12/51
B) With replacement: P(Both cards are clubs) = 13/52 * 13/52