A box of marbles contains an equal number 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.
red marbles : x
yellow marbles : x
green marbles : 2x
total 4x
prob(a red marble) = x/4x = 1/4
prob(a yellow marble) = x/4x = 1/4
prob(a green marble) = 2x/4x = 1/2
notice (1/4 + 1/4 + 1/2 = 1)
To assign probabilities to the elements, we need to determine the total number of marbles in the box and then calculate the probability for each color.
Let's denote:
- R: Red marbles
- Y: Yellow marbles
- G: Green marbles
From the given information, we know that:
- The number of red marbles is equal to the number of yellow marbles.
- The number of green marbles is twice the number of red marbles.
Let's assign a variable to the number of red marbles, say x.
Therefore, the number of yellow marbles would also be x, and the number of green marbles would be 2x.
To find the sum of all marbles, we add: x (red) + x (yellow) + 2x (green) = 4x.
Now, let's calculate the probabilities for each color:
Probability of drawing a red marble (P(R)):
The probability of drawing a red marble is equal to the number of red marbles over the total number of marbles:
P(R) = x/4x = 1/4.
Probability of drawing a yellow marble (P(Y)):
The probability of drawing a yellow marble is also equal to the number of yellow marbles over the total number of marbles:
P(Y) = x/4x = 1/4.
Probability of drawing a green marble (P(G)):
The probability of drawing a green marble is equal to the number of green marbles over the total number of marbles:
P(G) = 2x/4x = 1/2.
Note that the sum of all probabilities should be equal to 1, which indicates the entire sample space.
Therefore, the assigned probabilities are:
P(R) = 1/4
P(Y) = 1/4
P(G) = 1/2