Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
Let's walk through the steps to solve both parts of the question about drawing two jacks from a standard deck of 52 cards.
### Part (a): Drawing with Replacement
In this scenario, each time a card is drawn, it is placed back into the deck, and the deck is then reshuffled. This means that the probability of drawing a jack remains the same for both draws.
1. Determine the probability of drawing a jack in the first draw:
- There are 4 jacks in a standard deck of 52 cards.
- Therefore, the probability of drawing a jack is [tex]\( \frac{4}{52} \)[/tex].
2. Determine the probability of drawing a jack in the second draw after replacement:
- Because the card is replaced and the deck is reshuffled, the probability remains [tex]\( \frac{4}{52} \)[/tex].
3. Calculate the combined probability:
- Since these are independent events (due to replacement), the combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both Jacks}) = \left( \frac{4}{52} \right) \times \left( \frac{4}{52} \right) = \left( \frac{1}{13} \right) \times \left( \frac{1}{13} \right) = \frac{1}{169} \][/tex]
4. Convert to a decimal and round to the nearest ten-thousandth:
- [tex]\[ \frac{1}{169} \approx 0.0059 \][/tex]
So, the probability that both cards drawn are jacks with replacement is 0.0059.
### Part (b): Drawing without Replacement
In this scenario, once a card is drawn, it is not placed back into the deck, which affects the probabilities for the subsequent draws.
1. Determine the probability of drawing a jack in the first draw:
- There are still 4 jacks in a deck of 52 cards.
- Therefore, the probability of drawing a jack is [tex]\( \frac{4}{52} \)[/tex].
2. Determine the probability of drawing a jack in the second draw without replacement:
- After one jack is drawn, only 3 jacks are left in a reduced deck of 51 cards.
- So, the probability of drawing a jack in the second draw is [tex]\( \frac{3}{51} \)[/tex].
3. Calculate the combined probability:
- Since these are dependent events (due to no replacement), the combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both Jacks}) = \left( \frac{4}{52} \right) \times \left( \frac{3}{51} \right) = \left( \frac{1}{13} \right) \times \left( \frac{1}{17} \right) = \frac{1}{221} \][/tex]
4. Convert to a decimal and round to the nearest ten-thousandth:
- [tex]\[ \frac{1}{221} \approx 0.0045 \][/tex]
So, the probability that both cards drawn are jacks without replacement is 0.0045.
In summary:
- (a) The probability that both cards are jacks with replacement is 0.0059.
- (b) The probability that both cards are jacks without replacement is 0.0045.
### Part (a): Drawing with Replacement
In this scenario, each time a card is drawn, it is placed back into the deck, and the deck is then reshuffled. This means that the probability of drawing a jack remains the same for both draws.
1. Determine the probability of drawing a jack in the first draw:
- There are 4 jacks in a standard deck of 52 cards.
- Therefore, the probability of drawing a jack is [tex]\( \frac{4}{52} \)[/tex].
2. Determine the probability of drawing a jack in the second draw after replacement:
- Because the card is replaced and the deck is reshuffled, the probability remains [tex]\( \frac{4}{52} \)[/tex].
3. Calculate the combined probability:
- Since these are independent events (due to replacement), the combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both Jacks}) = \left( \frac{4}{52} \right) \times \left( \frac{4}{52} \right) = \left( \frac{1}{13} \right) \times \left( \frac{1}{13} \right) = \frac{1}{169} \][/tex]
4. Convert to a decimal and round to the nearest ten-thousandth:
- [tex]\[ \frac{1}{169} \approx 0.0059 \][/tex]
So, the probability that both cards drawn are jacks with replacement is 0.0059.
### Part (b): Drawing without Replacement
In this scenario, once a card is drawn, it is not placed back into the deck, which affects the probabilities for the subsequent draws.
1. Determine the probability of drawing a jack in the first draw:
- There are still 4 jacks in a deck of 52 cards.
- Therefore, the probability of drawing a jack is [tex]\( \frac{4}{52} \)[/tex].
2. Determine the probability of drawing a jack in the second draw without replacement:
- After one jack is drawn, only 3 jacks are left in a reduced deck of 51 cards.
- So, the probability of drawing a jack in the second draw is [tex]\( \frac{3}{51} \)[/tex].
3. Calculate the combined probability:
- Since these are dependent events (due to no replacement), the combined probability is the product of the individual probabilities:
[tex]\[ P(\text{Both Jacks}) = \left( \frac{4}{52} \right) \times \left( \frac{3}{51} \right) = \left( \frac{1}{13} \right) \times \left( \frac{1}{17} \right) = \frac{1}{221} \][/tex]
4. Convert to a decimal and round to the nearest ten-thousandth:
- [tex]\[ \frac{1}{221} \approx 0.0045 \][/tex]
So, the probability that both cards drawn are jacks without replacement is 0.0045.
In summary:
- (a) The probability that both cards are jacks with replacement is 0.0059.
- (b) The probability that both cards are jacks without replacement is 0.0045.
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
