A drawer contains n socks. When two are drawn randomly without replacement, the probability that both are red is 5/14. What is the smallest possible value of n

suppose there are x red socks among the n socks

so the prob of getting two reds
= (x/n)(x-1)/(n-1)
= x(x-1)/((n(n-1))

that is, numerator of our fraction consists of consecutive number multiples, and so does the denominator
... so it looks like 5/14 was reduced to lowest terms

let's build it up again, but looking at multiples of 5
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 ...
and those of 14
14 28 56 70 84 98 112 ....
5/14 = 10/28 or 15/42 or 20/56

ahh, 20/56 = (5x4)/(8x7)

So if there had been 8 socks of which 5 are red
prob(2red) = (5/8)(4/7) = 20/56 = 5/14

minimum value of n is 8

Let's start by setting up the problem.

The probability of drawing a red sock on the first draw is the number of red socks divided by the total number of socks. Since we are not replacing the sock after the first draw, the probability of drawing a red sock on the second draw depends on the number of red socks and the number of socks left in the drawer.

Let's say there are r red socks and n total socks in the drawer. The probability of drawing a red sock on the first draw is r/n, and if we don't replace the sock, the probability of drawing another red sock on the second draw is (r-1)/(n-1), since there is now one less sock in the drawer.

According to the problem, the probability of both socks being red is 5/14. So, we can write the following equation:

(r/n) * (r-1)/(n-1) = 5/14

Simplifying the equation, we get:

(r*(r-1))/(n*(n-1)) = 5/14

To find the smallest possible value of n, we need to find the smallest possible value of n that satisfies this equation.

Since n represents the total number of socks, it should be a positive integer. Also, since the probability is always less than or equal to 1, the numerator (r*(r-1)) should be less than or equal to (n*(n-1)).

To minimize n, we need to minimize the denominator (n*(n-1)). The smallest possible value for this denominator is when n = 2, which gives us a denominator of 2*1 = 2.

So, we can rewrite the equation using n = 2:

(r*(r-1))/2 = 5/14

Simplifying further, we get:

(r^2 - r)/2 = 5/14

Multiplying both sides of the equation by 2, we get:

r^2 - r = (5/14) * 2

r^2 - r = 10/14

r^2 - r = 5/7

To find the smallest possible value of r that satisfies this equation, we need to find the smallest positive integer value of r.

The smallest possible value for r is 1, which satisfies the equation:

1^2 - 1 = 0 = 5/7

Therefore, the smallest possible value of n is 2, which corresponds to having 1 red sock in the drawer.

To find the smallest possible value of n, we need to determine the number of red socks in the drawer.

Let's assume that the total number of socks in the drawer is n. Since we are drawing two socks without replacement, the first sock we draw has a probability of (n - 1)/n of not being replaced by a red sock.

If the first sock is not red, then there are (n - 1) socks remaining in the drawer, out of which we need to choose a red sock.

So, the probability of drawing two red socks is given by:
(n - 1)/n * (red socks remaining)/(total socks remaining)

From the given information, we have:
(n - 1)/n * (red socks remaining)/(total socks remaining) = 5/14

Simplifying the equation, we get:
14(n - 1)(red socks remaining) = 5n(total socks remaining)

Expanding and simplifying further, we can write:
14n - 14 = 5n(red socks remaining)
14n = 5n(red socks remaining) + 14

Since the number of red socks should be less than or equal to the total number of socks, we know that red socks remaining <= n. Thus, replacing the red socks remaining with n in the equation, we have:
14n = 5n(n) + 14

Simplifying the equation further, we get:
14n = 5n^2 + 14

Rearranging the equation, we have:
5n^2 - 14n + 14 = 0

To find the smallest possible value of n, we need to solve this quadratic equation. However, we can notice that this quadratic equation has no real solutions because the discriminant (b^2 - 4ac) is negative.

Therefore, there is no smallest possible value of n that satisfies the given conditions.