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Select the correct answer.

You are throwing darts at a dartboard. You have a [tex]\frac{1}{6}[/tex] chance of striking the bull's-eye each time you throw. If you throw 3 times, what is the probability that you will strike the bull's-eye all 3 times?

A. [tex]\frac{5}{136}[/tex]
B. [tex]\frac{1}{210}[/tex]
C. [tex]\frac{3}{233}[/tex]
D. [tex]\frac{1}{100}[/tex]


Sagot :

To determine the probability of striking the bull's-eye 3 times in a row when you have a [tex]\(\frac{1}{6}\)[/tex] chance to hit it each time, follow these steps:

1. Identify the probability of hitting the bull's-eye in a single throw:
The probability of hitting the bull's-eye with one throw is:
[tex]\[ \text{Probability} (\text{Single Throw}) = \frac{1}{6} \][/tex]

2. Calculate the probability of hitting the bull's-eye 3 times consecutively:
Since each throw is an independent event, the probability of hitting the bull's-eye 3 times in a row is the product of the probabilities of each individual throw.
[tex]\[ \text{Probability} (\text{3 Throws}) = \left( \frac{1}{6} \right) \times \left( \frac{1}{6} \right) \times \left( \frac{1}{6} \right) \][/tex]
Simplify this:
[tex]\[ \left( \frac{1}{6} \right)^3 = \frac{1}{216} \][/tex]

3. Compare the calculated probability with the given choices:
The calculated probability of hitting the bull's-eye 3 times in a row is [tex]\(\frac{1}{216}\)[/tex]. Now, compare this with the answer choices provided:
- A. [tex]\(\frac{5}{136} \approx 0.0367647\)[/tex]
- B. [tex]\(\frac{1}{210} \approx 0.0047619\)[/tex]
- C. [tex]\(\frac{3}{233} \approx 0.0128756\)[/tex]
- D. [tex]\(\frac{1}{100} = 0.01\)[/tex]

4. Identify the correct choice:
The choice that matches [tex]\(\frac{1}{216}\)[/tex] most closely is:
[tex]\[ \text{Choice B:} \frac{1}{210} \approx 0.0047619 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{B. \frac{1}{210}} \][/tex]