A bag holds 8 yellow marbles and 5 orange marbles. How many and what color marbles would you add to the bag so that the probability of picking a yellow marble is 2/3?
lets add x yellow marbles
so (x+8)/(x+13) = 2/3
3x + 24 = 2x + 26
x = 2
must add 2 yellows
or
lets add y orange marbles
8/(y+13) = 2/3
2y + 26 = 24
2y = -2
y = -1
or we could take out 1 orange marble
Proof:
yellow = 8
orange = 4
total = 12
prob of yellow = 8/12 = 2/3
To solve this problem, we need to find the number and color of marbles to add to the bag in order to make the probability of picking a yellow marble 2/3.
Let's start by calculating the initial probability of picking a yellow marble from the bag. We have a total of 8 yellow marbles, and 5 orange marbles, so the total number of marbles is 8 + 5 = 13.
The probability of picking a yellow marble is given by:
Probability(Yellow) = Number of Yellow Marbles / Total Number of Marbles
Since we want this probability to be 2/3, we can write the equation:
2/3 = Number of Yellow Marbles / (13 + Number of Marbles to Add)
Next, we can cross multiply and solve for the number of yellow marbles:
2(13 + Number of Marbles to Add) = 3(Number of Yellow Marbles)
Expanding the equation, we get:
26 + 2(Number of Marbles to Add) = 3(Number of Yellow Marbles)
Now we need to consider the number of marbles to add. Since we want to add marbles in order to increase the probability of picking a yellow marble, we can assume that the number of marbles to add must be greater than zero. Let's denote the number of marbles to add as "x".
Using this information, we can rewrite the equation as:
26 + 2x = 3(Number of Yellow Marbles)
Now, we need to find a solution that satisfies this equation, as well as ensures that the number of yellow marbles added to the total number of yellow marbles in the bag remains an integer.
To find a valid solution, we can start by substituting different values for "Number of Yellow Marbles" and "x" to see if any valid solutions exist.
For example, let's try substituting "Number of Yellow Marbles" = 8 (the initial number of yellow marbles). Plugging in these values, we get:
26 + 2x = 3(8)
26 + 2x = 24
2x = -2
x = -1
However, since we are looking for a positive value for "x" (the number of marbles to add), this solution is not valid.
We can continue substituting different values for "Number of Yellow Marbles" to check for valid solutions. However, after exploring various possibilities, we realize that there is no valid solution that satisfies both the equation and the requirement of adding marbles.
Therefore, it is not possible to add a specific number and color of marbles to the bag in order to make the probability of picking a yellow marble 2/3.