Suppose a box of marbles contains equal numbers of red marbles and yellow marbles but twice as many green marbles as red marbles. Draw one marble from the box and observe its color. Assign probabilities to the elements in the sample space. (Give your answers as fractions.)
To assign probabilities to the elements in the sample space, we need to first determine the total number of marbles in the box.
Let's denote the number of red marbles as "R", the number of yellow marbles as "Y", and the number of green marbles as "G".
Given that the box contains equal numbers of red and yellow marbles, we can represent this as: R = Y.
It is also given that there are twice as many green marbles as red marbles, so we can represent this as: G = 2R.
To find the total number of marbles, we can sum the number of each color: Total number of marbles = R + Y + G.
Since we know that R = Y, we can rewrite the equation: Total number of marbles = R + R + 2R.
Simplifying the equation, we get: Total number of marbles = 4R.
Now, let's assign probabilities to each element in the sample space based on the number of marbles of each color.
Element 1: Red marble (R)
Based on the information given, the probability of drawing a red marble can be calculated as: Number of red marbles / Total number of marbles = R / (R + R + 2R).
Simplifying, we get the probability of a red marble as: R / 4R = 1/4.
Element 2: Yellow marble (Y)
Since the box contains an equal number of red and yellow marbles, the probability of drawing a yellow marble is also 1/4.
Element 3: Green marble (G)
According to the information provided, there are twice as many green marbles as red marbles. So, the probability of drawing a green marble can be calculated as: Number of green marbles / Total number of marbles = 2R / (R + R + 2R).
Simplifying, we get the probability of a green marble as: 2R / 4R = 1/2.
Therefore, the probabilities for each element in the sample space are:
- Probability of a red marble: 1/4
- Probability of a yellow marble: 1/4
- Probability of a green marble: 1/2