Question

An urn has 144 marbles, orange and blue. A simple random sample of 2 marbles is taken from this urn. You are told that

• there are more orange than blue marbles in the urn.

• it is equally likely that the two marbles in the sample be of the same color as it is to to be of different colors.

What is the number of orange marbles in the urn?

Answer 1

Answer 2

Let's solve this problem step by step:

Let's assume the number of orange marbles in the urn is x. Therefore, the number of blue marbles would be 144 - x (since there are a total of 144 marbles in the urn).

According to the given information:

1. There are more orange marbles than blue marbles in the urn. This implies that x > (144 - x).

2. It is equally likely that the two marbles in the sample are of the same color as it is to be of different colors. This implies that the probability of drawing two orange marbles plus the probability of drawing two blue marbles is equal to the probability of drawing one orange and one blue marble.

Now let's calculate these probabilities:

The probability of drawing two orange marbles is given by: P(Orange, Orange) = (x/144) * ((x-1)/(144-1))

The probability of drawing two blue marbles is given by: P(Blue, Blue) = ((144-x)/144) * ((144-x-1)/(144-1))

The probability of drawing one orange and one blue marble is given by: P(Orange, Blue) + P(Blue, Orange) = 2 * (x/144) * ((144-x)/(144-1))

According to the given information, the probability of two marbles being the same color is equal to the probability of two marbles being different colors:

P(Orange, Orange) + P(Blue, Blue) = P(Orange, Blue) + P(Blue, Orange)

Substituting the calculated probabilities:

(x/144) * ((x-1)/(144-1)) + ((144-x)/144) * ((144-x-1)/(144-1)) = 2 * (x/144) * ((144-x)/(144-1))

Simplifying and solving for x:

x(x-1) + (144-x)(144-x-1) = 2x(144-x) x^2 - x + 144^2 - 145x + x^2 = 288x - 2x^2 2x^2 - 289x + 144^2 = 0

Solving this quadratic equation using the quadratic formula, we find:

x = (289 ± √(289^2 - 42144^2))/(2*2)

Calculating the value of x using a calculator, we get:

x ≈ 131.531 or x ≈ 112.469

Since x represents the number of orange marbles, it cannot be a decimal value. Therefore, the number of orange marbles in the urn is approximately 112.