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A game consists of spinning an arrow around a stationary disk as shown below. When the arrow comes to rest, there are eight equally likely outcomes. It could come to rest in any one of the sectors numbered $1, 2, 3,4, 5, 6, 7$ or $8$ as shown.

Two such disks are used in a game where their arrows are independently spun.

What is the probability that the sum of the numbers on the resulting sectors upon spinning the two disks is equal to 8 after the arrows come to rest?

  1. $\dfrac{1}{16}$
  1. $\dfrac{5}{64}$
  1. $\dfrac{3}{32}$
  1. $\dfrac{7}{64}$
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Given the, two disks

When we select only one number from any of the two disks, then the probability $ = \frac{1}{8}$

When we select one number from the first disk and the one number from the second disk simultaneously, then the probability $ = \frac{1}{8} \times \frac{1}{8} = \frac{1}{64}$

 The numbers on the resulting sectors upon spinning the two disks is equal to $8$ after the arrows come to rest $ = \underbrace{(1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1)}_{\text{Number of favorable outcomes = 7}}$

$\therefore$ The required probability $ = \dfrac{7}{64}$

Correct Answer $:\text{D}$

${\color{Magenta}{\textbf{PS:}}}$ In probability, two events are independent if the incidence of one event does not affect the probability of the other event. If the incidence of one event does affect the probability of the other event, then the events are dependent.

If two events $\text{A},$ and $\text{B}$ are independent, then ${\color{Green}{\boxed{\text{P(A} \cap \text{B}) = \text{P(A)} \cdot \text{P(B)}}}}$

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