Question
You have a deck of cards that are numbered 1-12 and you have a regular 6 sided cube.
A- Find the probability of choosing a 6 from the deck of cards and getting an even number on the die.
B- Find the probability of choosing a number greater than 5 from the deck of cards and getting a number less then 3 on the die.
A- Find the probability of choosing a 6 from the deck of cards and getting an even number on the die.
B- Find the probability of choosing a number greater than 5 from the deck of cards and getting a number less then 3 on the die.
Answers
GPT-4o mini
To find the probabilities for each part, we can use the formula for probability, which is:
\[
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
### Part A: Find the probability of choosing a 6 from the deck of cards and getting an even number on the die.
1. **Total outcomes for the deck of cards**: There are 12 cards, numbered from 1 to 12.
2. **Favorable outcomes for choosing a 6**: There is only 1 "6" in the deck.
3. **Total outcomes for the die**: A regular 6-sided die has numbers 1 to 6 (total 6 outcomes).
4. **Favorable outcomes for getting an even number on the die**: The even numbers on the die are 2, 4, and 6, which gives us 3 favorable outcomes.
Now, let's calculate the probability for both the card and the die:
- The probability of choosing a 6 from the deck:
\[
P(\text{choosing a 6}) = \frac{1}{12}
\]
- The probability of getting an even number on the die:
\[
P(\text{even number on the die}) = \frac{3}{6} = \frac{1}{2}
\]
Since these two events are independent (the choice of card does not affect the die roll), we multiply the probabilities:
\[
P(A) = P(\text{choosing a 6}) \times P(\text{even number on the die}) = \frac{1}{12} \times \frac{1}{2} = \frac{1}{24}
\]
### Part B: Find the probability of choosing a number greater than 5 from the deck of cards and getting a number less than 3 on the die.
1. **Favorable outcomes for choosing a number greater than 5**: The numbers greater than 5 in the deck are 6, 7, 8, 9, 10, 11, and 12. This gives us 7 favorable outcomes.
2. **Total outcomes for the deck of cards**: Again, there are 12 cards.
3. **Total outcomes for the die**: A regular 6-sided die has numbers 1 to 6 (total 6 outcomes).
4. **Favorable outcomes for getting a number less than 3 on the die**: The numbers less than 3 on the die are 1 and 2, which gives us 2 favorable outcomes.
Now we will calculate the probabilities for both the card and the die:
- The probability of choosing a number greater than 5 from the deck:
\[
P(\text{choosing a number > 5}) = \frac{7}{12}
\]
- The probability of getting a number less than 3 on the die:
\[
P(\text{number < 3 on the die}) = \frac{2}{6} = \frac{1}{3}
\]
Since these two events are independent, we multiply the probabilities:
\[
P(B) = P(\text{choosing a number > 5}) \times P(\text{number < 3 on the die}) = \frac{7}{12} \times \frac{1}{3} = \frac{7}{36}
\]
### Summary of Results:
- **Part A**: Probability of choosing a 6 from the deck and getting an even number on the die \( P(A) = \frac{1}{24} \)
- **Part B**: Probability of choosing a number greater than 5 from the deck and getting a number less than 3 on the die \( P(B) = \frac{7}{36} \)
\[
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
### Part A: Find the probability of choosing a 6 from the deck of cards and getting an even number on the die.
1. **Total outcomes for the deck of cards**: There are 12 cards, numbered from 1 to 12.
2. **Favorable outcomes for choosing a 6**: There is only 1 "6" in the deck.
3. **Total outcomes for the die**: A regular 6-sided die has numbers 1 to 6 (total 6 outcomes).
4. **Favorable outcomes for getting an even number on the die**: The even numbers on the die are 2, 4, and 6, which gives us 3 favorable outcomes.
Now, let's calculate the probability for both the card and the die:
- The probability of choosing a 6 from the deck:
\[
P(\text{choosing a 6}) = \frac{1}{12}
\]
- The probability of getting an even number on the die:
\[
P(\text{even number on the die}) = \frac{3}{6} = \frac{1}{2}
\]
Since these two events are independent (the choice of card does not affect the die roll), we multiply the probabilities:
\[
P(A) = P(\text{choosing a 6}) \times P(\text{even number on the die}) = \frac{1}{12} \times \frac{1}{2} = \frac{1}{24}
\]
### Part B: Find the probability of choosing a number greater than 5 from the deck of cards and getting a number less than 3 on the die.
1. **Favorable outcomes for choosing a number greater than 5**: The numbers greater than 5 in the deck are 6, 7, 8, 9, 10, 11, and 12. This gives us 7 favorable outcomes.
2. **Total outcomes for the deck of cards**: Again, there are 12 cards.
3. **Total outcomes for the die**: A regular 6-sided die has numbers 1 to 6 (total 6 outcomes).
4. **Favorable outcomes for getting a number less than 3 on the die**: The numbers less than 3 on the die are 1 and 2, which gives us 2 favorable outcomes.
Now we will calculate the probabilities for both the card and the die:
- The probability of choosing a number greater than 5 from the deck:
\[
P(\text{choosing a number > 5}) = \frac{7}{12}
\]
- The probability of getting a number less than 3 on the die:
\[
P(\text{number < 3 on the die}) = \frac{2}{6} = \frac{1}{3}
\]
Since these two events are independent, we multiply the probabilities:
\[
P(B) = P(\text{choosing a number > 5}) \times P(\text{number < 3 on the die}) = \frac{7}{12} \times \frac{1}{3} = \frac{7}{36}
\]
### Summary of Results:
- **Part A**: Probability of choosing a 6 from the deck and getting an even number on the die \( P(A) = \frac{1}{24} \)
- **Part B**: Probability of choosing a number greater than 5 from the deck and getting a number less than 3 on the die \( P(B) = \frac{7}{36} \)