Probability of lights burning out in Series Circuit

In summary, The probability of a string of Christmas lights failing in a 3-year period is not simply the sum of the individual probabilities of each light failing, as each light failing does not guarantee the entire string failing. Instead, the probability of the string failing is calculated by finding the probability that none of the lights fail, which is 0.95 for each light, and then subtracting that from 1 to get the probability that at least one light will fail. This results in a probability of 0.18549375, which is significantly lower than the initial answer of 0.20. Additionally, it is possible for more than one light to fail at the same time in a series circuit.
  • #1
lth2525
2
0
Hi, i have this question wrong in my intro statistics class and I'm wonder if someone could help me understand why. Thanks in advance.

Heres the question:

"A string of christmas lights contains 4 bulbs. The lights are wired in a series, so that if one fails, the whole sting would go dark. Each light has a probability of 0.05 of failing in a 3-year period. The lights fail independently of each other. What is the probability that a sting of lights will fail in a 3 year period?"

Now, my answer was 0.20, because each lights fail independetnly of each other, so then, if one fails, the others do not get a chance to fail. Since logically, w/o any external influence, a light bulb should not fail if it is not lit, therefore, 4 x .05 = 0.20. But i got the asnwer wrong, can someone help me to understand if I'm missing something? Thanks

On another note, can more than one light in a series circuit burn out at the same time?
 
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  • #2
You can't add probabilities like that. Think about it, if you had 4 bulbs and each had a 0.25 probability of failing, you would get a 1 (100%) chance of failing for an answer and it is clear that this is not correct because it can still happen that none of the four fail. If you throw two coins, the chance that each will turn heads is 50%, but it's not 100% that at least one will :wink:

Try to think about it in a different way. What is the chance that a single lightbulb won't fail? Then, what is the chance that NONE of the bulbs will fail?
 
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  • #3
The probability of anyone bulb failing is 0.05 so the probability that anyone bulb WON'T fail is 0.95. The probability that NONE of the four lights will fail is (0.95)4= 0.81450625. Since the light string will fail is any of them does, that is the probability that the string will NOT fail so the probability that it will is 1-0.81450625= 0.18549375.
 
  • #4
O, Ok, thanks for explaining that to me, i understand it now
 

1. What is a series circuit?

A series circuit is a type of electrical circuit in which the components are connected in a single loop or pathway. This means that the current flows through each component in the same direction.

2. How does the probability of lights burning out differ in a series circuit compared to a parallel circuit?

In a series circuit, the probability of lights burning out is higher compared to a parallel circuit. This is because in a series circuit, the same amount of current flows through each component, so if one component burns out, the entire circuit is broken and all the lights will go out. In a parallel circuit, the current is divided among the different branches, so if one component burns out, the remaining branches can still function.

3. What factors can affect the probability of lights burning out in a series circuit?

The main factor that affects the probability of lights burning out in a series circuit is the quality and durability of the components. If the components are of high quality and can handle the flow of current, the probability of burning out is lower. Other factors such as temperature, voltage, and age of the components can also play a role.

4. Can the probability of lights burning out in a series circuit be calculated?

Yes, the probability of lights burning out in a series circuit can be calculated using the formula: P = 1 - (1 - p)^n, where P is the probability of all lights burning out, p is the probability of one light burning out, and n is the number of lights in the circuit. However, this calculation assumes that all components have an equal chance of burning out.

5. How can the probability of lights burning out in a series circuit be reduced?

The probability of lights burning out in a series circuit can be reduced by using high-quality and durable components, properly maintaining the circuit, and ensuring that the circuit is not overloaded with too much current. Adding fuses or circuit breakers can also help protect the components and reduce the probability of burning out.

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