A box contains n balls of which 2 are red and the rest are green. Two balls are to be selected at random one after the other from the box without replacement. If the probability of selecting 2 red balls is 1/28, find;
the value of n
the probability that a red is selected followed by a green
for both red, we need
2/(n-2) * 1/(n-1) = 1/28
n = 8
Now you can do the other part.
To find the value of n, we can use the concept of probability. The probability of selecting 2 red balls out of n balls can be expressed as:
P(2 red balls) = (number of ways to choose 2 red balls) / (number of ways to choose 2 balls from n balls without replacement)
The number of ways to choose 2 red balls out of 2 is 1, and the number of ways to choose 2 balls from n balls without replacement can be calculated using the combination formula: C(n, 2) = n! / (2!(n-2)!).
Therefore, the probability of selecting 2 red balls is 1/n(n-1) = 1/28.
To solve for n, we can set up the equation:
1/n(n-1) = 1/28
Cross-multiplying, we get:
28n(n-1) = 1
Expanding, we have:
28n^2 - 28n - 1 = 0
Using the quadratic formula, we find:
n = (28 ± sqrt(28^2 - 4 * 28 * (-1))) / (2 * 28)
Simplifying further, we get:
n = (28 ± sqrt(784 + 112)) / 56
n = (28 ± sqrt(896)) / 56
n = (28 ± 28sqrt(2)) / 56
n = (1 ± sqrt(2)) / 2
Since n represents the number of balls, it cannot be negative. Therefore, the value of n is:
n = (1 + sqrt(2)) / 2
To find the probability that a red ball is selected followed by a green ball, we need to account for the two possible cases: the first ball is red and the second is green, and the first ball is green and the second is red.
Case 1: Red ball followed by green ball
The probability of selecting a red ball first is 2/n (since there are 2 red balls out of n total), and the probability of selecting a green ball next is (n-2)/(n-1) (since there are n-2 green balls remaining out of n-1 total after the first selection).
Therefore, the probability of selecting a red ball followed by a green ball is:
P(Red followed by Green) = (2/n) * ((n-2)/(n-1))
Simplifying, we have:
P(Red followed by Green) = 2(n-2)/(n(n-1))
You can substitute the value of n into this equation to find the exact probability.
To find the value of n, we can use the information given in the problem. We know that there are 2 red balls and the rest are green. Therefore, the total number of balls in the box can be calculated as:
n = Number of red balls + Number of green balls
n = 2 + (the rest)
Next, we can calculate the probability of selecting 2 red balls using the concept of combination. Since we are selecting 2 balls without replacement, the probability can be calculated as:
P(2 red) = (Number of ways to select 2 red balls) / (Number of ways to select 2 balls)
The number of ways to select 2 red balls can be calculated as the combination of 2 red balls out of the total number of red balls:
Number of ways to select 2 red balls = C(2, Number of red balls)
Similarly, the number of ways to select any 2 balls can be calculated as the combination of 2 balls out of the total number of balls:
Number of ways to select 2 balls = C(2, n)
Substituting the above values into the probability formula:
P(2 red) = C(2, Number of red balls) / C(2, n)
Given that P(2 red) = 1/28, we can write the equation as:
1/28 = C(2, 2) / C(2, n)
Simplifying this equation:
1/28 = 1 / C(2, n)
Now, we can find the value of n by solving this equation. By analysing the possible values for n, we can determine that the value of n is 8.
To find the probability that a red ball is selected first followed by a green ball, we need to consider the two possible combinations: RG and GR.
Number of ways to select RG = Number of red balls × Number of green balls = 2 × (n - 2) = 2(n - 2)
Number of ways to select GR = Number of green balls × Number of red balls = (n - 2) × 2 = 2(n - 2)
The total number of ways to select any 2 balls remains the same: C(2, n).
Therefore, the probability of selecting a red ball first followed by a green ball is:
P(RG or GR) = (Number of ways to select RG + Number of ways to select GR) / (Number of ways to select 2 balls)
P(RG or GR) = (2(n - 2) + 2(n - 2)) / C(2, n)
Simplifying this equation gives us the answer.