. A bag contains 3 marbles: a red one, a white one, and a green one. Suppose that two marbles are selected without replacement.
(a) How many outcomes are in the sample space? List them
(b) What is the probability that a red marble, and then a green marble (in that order) are selected?
(a) There are 3 marbles in the bag, so for the first pick, you have 3 options. After one marble is chosen, there are 2 marbles left in the bag for the second pick. So, there are a total of 3 * 2 = 6 outcomes in the sample space.
The sample space can be listed as:
1. Red, White
2. Red, Green
3. White, Red
4. White, Green
5. Green, Red
6. Green, White
(b) The probability of selecting a red marble and then a green marble (in that order) can be calculated as follows:
For the first pick, there is 1 red marble out of 3 marbles, so the probability is 1/3.
After one marble is chosen, there is 1 green marble left out of 2 marbles, so the probability is 1/2.
To find the probability of both events occurring, we multiply the probabilities: (1/3) * (1/2) = 1/6.
So, the probability that a red marble and then a green marble are selected is 1/6, which can also be written as "Highly improbable, but you might get lucky!"
(a) To find the number of outcomes in the sample space, we can use the concept of combinations. Since we are selecting two marbles without replacement, the order of selection does not matter.
We have 3 marbles to choose from, and we are selecting 2 marbles. Therefore, the number of outcomes in the sample space can be calculated using the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n is 3 (the total number of marbles) and r is 2 (the number of marbles being selected). Plugging these values into the formula:
3C2 = 3! / (2!(3-2)!)
= 3! / (2! * 1!)
= (3 * 2 * 1) / ((2 * 1) * 1)
= 3
Therefore, there are 3 outcomes in the sample space. Let's list them:
1. Red, White
2. Red, Green
3. White, Green
(b) To find the probability that a red marble, and then a green marble (in that order) are selected, we need to find the favorable outcome and divide it by the total number of outcomes in the sample space.
In this case, there is only one favorable outcome: Red, Green. And as we calculated earlier, there are 3 outcomes in the sample space.
Therefore, the probability can be calculated as:
Probability = Favorable Outcomes / Total Outcomes
= 1 / 3
So, the probability that a red marble, and then a green marble (in that order) are selected is 1/3.
To find the number of outcomes in the sample space, we need to consider all the possible ways of selecting two marbles without replacement. Let's analyze each question separately:
(a) How many outcomes are in the sample space? List them
To find the number of outcomes in the sample space, we use the concept of combinations. Since we are selecting two marbles from a bag of three without replacement, the formula for calculating combinations is nCr, where n is the total number of marbles and r is the number of marbles to be selected.
In this case, n = 3 (as there are 3 marbles in the bag) and r = 2 (as we are selecting 2 marbles at a time). So, the number of outcomes in the sample space is given by:
3C2 = 3! / (2!(3-2)!) = 3
The three possible outcomes are:
1. Red, White
2. Red, Green
3. White, Green
(b) What is the probability that a red marble and then a green marble (in that order) are selected?
Since we need to select the red marble first and then the green marble, there is only one outcome that satisfies this condition: Red, Green.
The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Here, the total number of outcomes is 3 (as calculated in part (a)), and there is only one favorable outcome.
Therefore, the probability of selecting a red marble and then a green marble (in that order) is 1/3.
sample sapce
r w
r g
w r
w g
g r
g w
You have 1 chance out of 6 to get rg in that order.