Answer:
Explanation:
a) If we denote the number of red counters as R and the number of blue counters as B, then the probability of selecting one counter of each color is:
P(one red and one blue) = (R/(R+B)) * (B/(R+B-1)) = R * B / (R+B)(R+B-1)
We can simplify this expression by canceling the common factors of R and B:
P(one red and one blue) = (R * B) / (R+B)(R+B-1) = (R * B) / (R^2 + 2RB + B^2 - R - B)
Simplifying further, we get:
P(one red and one blue) = (R * B) / (R^2 - R + B^2 - B)
Since we want to express this probability in terms of n, we can substitute R + B = n to get:
P(one red and one blue) = (R * (n-R)) / (R^2 - R + (n-R)^2 - (n-R))
Simplifying again, we get:
P(one red and one blue) = (R * (n-R)) / (R^2 - 2Rn + n^2 + R)
Finally, we can simplify this expression to get:
P(one red and one blue) = (R * (n-R)) / (n(n-1))
This is the probability of selecting one red and one blue counter.
b) If the probability of selecting one red and one blue counter is 0.125, then we can substitute this value into the expression we derived above to solve for the number of red counters:
0.125 = (R * (n-R)) / (n(n-1))
Multiplying both sides by n(n-1) gives:
(0.125) * (n(n-1)) = R * (n-R)
Expanding the right side gives:
(0.125) * (n(n-1)) = R * n - R^2
Rearranging the terms gives:
R^2 - R*n + (0.125) * (n(n-1)) = 0
This is a quadratic equation in R, which we can solve using the quadratic formula:
R = (n +/- sqrt(n^2 - 4*(0.125)*(n(n-1)))) / 2
Substituting the value of (0.125) into the equation gives:
R = (n +/- sqrt(n^2 - 0.5n(n-1))) / 2
We want the positive solution for R, since R must be a positive integer. Therefore, the number of red counters is:
R = (n + sqrt(n^2 - 0.5n(n-1))) / 2
To find the integer value of R, we can round this expression to the nearest integer. For example, if n = 8, then the number of red counters is:
R = (8 + sqrt(8^2 - 0.587)) / 2 = 4.29... ~ 4
If we substitute this value of R back into the equation for the probability of selecting one red and one blue counter, we get:
P(one red and one blue) = (4 * (8-4)) / (8(8-1)) = 0.125
We can check that this value is indeed equal to the given probability of 0.125, so our solution is correct.
Note that we could also have solved for R by substituting the given probability of 0.125 directly into the equation for the probability of selecting one red and one blue counter and solving for R. This would give us the same result.