Respuesta :
By definition of a binomial random variable, the probability that the machine will be working is 0.82045.
Definition of probability
First of all, you should know that the Probability is the greater or lesser possibility that a certain event will occur.
In other words, the probability establishes a relationship between the number of favorable events and the total number of possible events.
Definition of random variable
On the other hand, a random variable is a function that associates a real number, perfectly defined, to each sample point.
A random variable is discrete if the numbers it gives rise to are integers.
Definition of binomial distribution
Finally, a binomial distribution is a discrete probability distribution that describes the number of successes in performing n independent experiments on a random variable.
The expression to calculate the binomial distribution is:
P(X=x)=([tex]n\\ x[/tex]) pˣ qⁿ⁻ˣ
Where:
- n = Number of trials/experiments
- x = Number of successes
- p = Probability of success
- q = Probability of failure (1-p)
Remember that:
([tex]n\\ x[/tex])=[tex]\frac{n!}{x!(n-x)!}[/tex]
where the exclamation mark represents the factorial symbol, that is, the product of all positive integers from 1 to n.
Probability that the machine will be working
A machine has 28 identical components which function independently. The probability that a component will fail is 0.19.
The following random variable is defined:
x= the number of components will stop working.
Being n the total number of pieces (in this case, 12 pieces) and p the probability of success (the piece fails, in this case 0.19), the expression to be used is expressed as:
P(X=x)=([tex]12\\ x[/tex]) 0.19ˣ (1-0.19)¹²⁻ˣ
P(X=x)=[tex]\frac{12!}{x!(12-x)!}[/tex] 0.19ˣ (0.81)¹²⁻ˣ
Since the machine will stop working if more than three components fail, you need to know the probability that the machine will be working if 3 or less than 3 components fail. In other words, it may be that 0, 1, 2 or 3 pieces are broken and the machine continues to work:
P(X≤3)= P(X=0) + P(X=1) + P(X=2) + P(X=3)
Then:
P(X≤3)= [tex]\frac{12!}{0!(12-0)!}[/tex] 0.19⁰ (0.81)¹²⁻⁰ + [tex]\frac{12!}{1!(12-1)!}[/tex] 0.19¹ (0.81)¹²⁻¹ + [tex]\frac{12!}{2!(12-2)!}[/tex] 0.19² (0.81)¹²⁻² + [tex]\frac{12!}{3!(12-3)!}[/tex] 0.19³ (0.81)¹²⁻³
Solving:
P(X≤3)= [tex]\frac{12!}{0!12!}[/tex] ×1×(0.81)¹² + [tex]\frac{12!}{1!11!}[/tex] 0.19×(0.81)¹¹ + [tex]\frac{12!}{2!10!}[/tex] 0.19²×(0.81)¹⁰ + [tex]\frac{12!}{3!9!}[/tex] 0.19³×(0.81)⁹
P(X≤3)= 1×1×(0.81)¹² + 12×0.19×(0.81)¹¹ + 66×0.19²×(0.81)¹⁰ + 220×0.19³×(0.81)⁹
P(X≤3)= (0.81)¹² + 2.28×(0.81)¹¹ + 2.3826×(0.81)¹⁰ + 1.50898×(0.81)⁹
P(X≤3)= 0.82045
Finally, the probability that the machine will be working is 0.82045.
Learn more about binomial random variable:
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https://brainly.com/question/14482254
https://brainly.com/question/14867806
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