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Difficulty: HardIndependent and Dependent Events

A box contains xx blue spheres and 88 yellow spheres, where xx is a positive integer. If two spheres are selected at random one after another without replacement, the probability that both selected spheres are blue is 517\frac{5}{17}. What is the total number of spheres in the box initially?

  1. A
    10
  2. B
    15
  3. 18Answer
  4. D
    20
  5. E
    25

Answer

The total number of spheres in the box initially is 18.
The probability of drawing two blue spheres sequentially without replacement is given by xx+8×x−1x+7=517\frac{x}{x+8} \times \frac{x-1}{x+7} = \frac{5}{17}. Expanding and rearranging the equation gives 3x2−23x−70=03x^2 - 23x - 70 = 0, which factors into (3x+7)(x−10)=0(3x + 7)(x - 10) = 0. Since xx must be a positive integer, x=10x = 10. The total number of spheres in the box initially is x+8=10+8=18x + 8 = 10 + 8 = 18.

Step-by-Step Solution

1
Set up the probability expression for dependent sequential events.
P(\text{both blue}) = \frac{x}{x+8} \times \frac{x-1}{x+7}
Because the draws occur without replacement, the total count decreases from x+8x+8 to x+7x+7 and the number of blue spheres decreases from xx to x−1x-1 for the second draw.
2
Equate to the given probability and clear denominators to form a quadratic equation.
\frac{x(x-1)}{(x+8)(x+7)} = \frac{5}{17} \implies 17(x^2 - x) = 5(x^2 + 15x + 56) \implies 3x^2 - 23x - 70 = 0
Cross-multiplying converts the rational probability equation into a standard quadratic equation.
3
Factor the quadratic equation to find the positive integer root.
(3x + 7)(x - 10) = 0 \implies x = 10
Since the count of spheres must be a positive integer, x=10x = 10 is the only valid solution for the number of blue spheres.
4
Calculate the total initial number of spheres.
\text{Total} = x + 8 = 10 + 8 = 18
The question asks for the total initial number of spheres, which includes both the xx blue spheres and the 8 yellow spheres.

Key Concept

Probability of Dependent Events Without Replacement
Estimated Time:2m 0s
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Topics

Independent and Dependent Events

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