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Zorluk: ZorProbability of Independent, Dependent, and Mutually Exclusive Events

A container holds nn spheres, exactly 5 of which are blue and the remaining n−5n - 5 are green. Two spheres are drawn at random from the container one after another without replacement. If the probability that at least one of the selected spheres is green is 1415\frac{14}{15}, what is the total number of spheres nn in the container?

  1. A
    15
  2. B
    20
  3. 25Cevap
  4. D
    30
  5. E
    75

Cevap

The total number of spheres nn in the container is 25.
The probability of at least one green sphere is complementary to drawing zero green spheres (meaning both spheres drawn are blue). Subtracting 1415\frac{14}{15} from 1 yields P(both blue)=115P(\text{both blue}) = \frac{1}{15}. Since the selection is without replacement, P(both blue)=5n×4n−1=20n(n−1)P(\text{both blue}) = \frac{5}{n} \times \frac{4}{n-1} = \frac{20}{n(n-1)}. Equating this to 115\frac{1}{15} gives n(n−1)=300n(n-1) = 300. Solving the quadratic equation n2−n−300=0n^2 - n - 300 = 0 gives n=25n = 25, which correctly represents the total number of spheres.

Adım Adım Çözüm

1
Use the complement rule to determine the probability that both selected spheres are blue.
P(both blue)=1−P(at least one green)=1−1415=115P(\text{both blue}) = 1 - P(\text{at least one green}) = 1 - \frac{14}{15} = \frac{1}{15}.
The event that at least one sphere is green is the complement of the event that both drawn spheres are blue.
2
Set up the joint probability equation for drawing two blue spheres sequentially without replacement.
P(both blue)=5n×4n−1=20n(n−1)P(\text{both blue}) = \frac{5}{n} \times \frac{4}{n - 1} = \frac{20}{n(n - 1)}.
There are 5 blue spheres initially out of nn. After drawing one blue sphere, 4 blue spheres remain out of n−1n - 1 total spheres.
3
Equate the expressions and solve for nn.
\begin{aligned} \frac{20}{n(n - 1)} &= \frac{1}{15} \\ n(n - 1) &= 300 \\ n^2 - n - 300 &= 0 \\ (n - 25)(n + 12) &= 0 \end{aligned}
Cross-multiplying gives a quadratic equation in terms of nn.
4
Select the valid positive integer solution for nn.
n=25n = 25 (since n>0n > 0).
The total number of spheres must be a positive integer.

Anahtar Kavram

Probability of Complementary Events and Dependent Sequential Events
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Konular

Probability of Independent, Dependent, and Mutually Exclusive Events

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