Suppose you have a bag containing two red marbles, two blue marbles, and two white marbles. You choose two marbles without looking.
a. What is the probability that you will choose a red marble and then a blue marble without replacing the red one?
b. What is the probability that you will choose two red marbles in a row without replacing the first one?
a) 2/6 picking the red and 2/5 picking the blue. (2/6)*(2/5) = 4/30
b) (2/6) picking the red (1/5) picking the second (2/6)*(1/5) = 2/30
a. Well, since there are two red marbles out of six total marbles, the probability of picking a red marble first is 2/6. After you've taken out the red marble, there will be five marbles left, one of which is blue. So the probability of picking a blue marble second is 1/5. To find the overall probability, you multiply the probabilities together: (2/6) * (1/5) = 1/15. So the probability of choosing a red marble and then a blue marble without replacement is 1/15.
b. If you want to choose two red marbles in a row without replacing the first one, then you first need to pick a red marble on your first attempt. We already know that the probability of choosing a red marble is 2/6. After you've picked the first red marble, there will be one red marble left, out of five remaining marbles. So the probability of picking another red marble is 1/5. Again, to find the overall probability, you multiply the probabilities together: (2/6) * (1/5) = 1/15. So the probability of choosing two red marbles in a row without replacing the first one is also 1/15.
a. To find the probability of choosing a red marble and then a blue marble without replacing the red one, we need to calculate the probability of each step and then multiply them together.
Step 1: Probability of choosing a red marble
Out of the total of 6 marbles, there are 2 red marbles. So, the probability of choosing a red marble on the first draw is 2/6.
Step 2: Probability of choosing a blue marble without replacing the red one
After choosing the red marble in the previous step, there will be 5 marbles left in the bag, with 2 of them being blue. So, the probability of choosing a blue marble without replacing the red one is 2/5.
To calculate the overall probability, we multiply the probabilities of each step:
P(red and then blue) = P(red) * P(blue without replacing the red)
P(red and then blue) = (2/6) * (2/5)
P(red and then blue) = 4/30
P(red and then blue) = 2/15
Therefore, the probability of choosing a red marble and then a blue marble without replacing the red one is 2/15.
b. To find the probability of choosing two red marbles in a row without replacing the first one, we again need to calculate the probability of each step and multiply them together.
Step 1: Probability of choosing a red marble
Out of the total of 6 marbles, there are 2 red marbles. So, the probability of choosing a red marble on the first draw is 2/6.
Step 2: Probability of choosing another red marble without replacing the first one
After choosing the first red marble, there will be 5 marbles left in the bag, with only 1 red marble remaining. So, the probability of choosing another red marble without replacing the first one is 1/5.
To calculate the overall probability, we multiply the probabilities of each step:
P(two reds in a row) = P(red) * P(red without replacing the first one)
P(two reds in a row) = (2/6) * (1/5)
P(two reds in a row) = 2/30
P(two reds in a row) = 1/15
Therefore, the probability of choosing two red marbles in a row without replacing the first one is 1/15.
To solve these probability problems, we need to find the number of favorable outcomes (i.e., the number of ways the desired event can occur) and the total number of possible outcomes.
a. Probability of choosing a red marble and then a blue marble without replacing the red one:
Step 1: Calculate the number of ways to choose a red marble and a blue marble.
- There are two red marbles, so we can choose one of them in 2 ways.
- Once we choose a red marble, there is one less red marble in the bag.
- There are two blue marbles remaining, so we can then choose one of them in 2 ways.
- Therefore, the number of ways to choose a red marble and a blue marble is 2 * 2 = 4.
Step 2: Calculate the total number of possible outcomes when choosing two marbles.
- Initially, there are six marbles in the bag.
- When we choose the first marble, there are six possibilities.
- After removing one marble, there remain five marbles in the bag.
- When we choose the second marble, there are five possibilities.
- Therefore, the total number of possible outcomes is 6 * 5 = 30.
Step 3: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
- Probability = Number of favorable outcomes / Total number of possible outcomes
- Probability = 4 / 30
- Probability = 2 / 15
Answer: The probability of choosing a red marble and then a blue marble without replacing the red one is 2/15.
b. Probability of choosing two red marbles in a row without replacing the first one:
Step 1: Calculate the number of ways to choose two red marbles.
- There are two red marbles, so we can choose one of them in 2 ways.
- Once we choose the first red marble, there is one less red marble in the bag.
- There is still one red marble remaining, so we can then choose it in 1 way.
- Therefore, the number of ways to choose two red marbles is 2 * 1 = 2.
Step 2: Calculate the total number of possible outcomes when choosing two marbles.
- Initially, there are six marbles in the bag.
- When we choose the first marble, there are six possibilities.
- After removing one marble, there remain five marbles in the bag.
- When we choose the second marble, there are five possibilities.
- Therefore, the total number of possible outcomes is 6 * 5 = 30.
Step 3: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes.
- Probability = Number of favorable outcomes / Total number of possible outcomes
- Probability = 2 / 30
- Probability = 1 / 15
Answer: The probability of choosing two red marbles in a row without replacing the first one is 1/15.