Respuesta :
Answer:
a) 18.77% probability that the driller drills at 10 locations and has 1 success
b) 75.60% probability that the driller drills at 10 locations and has at least 2 success
Step-by-step explanation:
For each drill, there are only two possible outcomes. Either it is a success, or it is not. The probability of a drill being a success is independent of other drills. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
10 locations
This means that [tex]n = 10[/tex]
Suppose the probability of a success at any specific location is 0.25.
This means that [tex]p = 0.25[/tex]
(a) What is the probability that the driller drills at 10 locations and has 1 success?
This is P(X = 1).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 1) = C_{10,1}.(0.25)^{1}.(0.75)^{9} = 0.1877[/tex]
18.77% probability that the driller drills at 10 locations and has 1 success
(b) What is the probability that the driller drills at 10 locations and has at least 2 success?
Either there are less than 2 success, or there are at least 2. The sum of the probabilities of these events is decimal 1. So
[tex]P(X < 2) + P(X \geq 2) = 1[/tex]
We want [tex]P(X \geq 2)[/tex]
So
[tex]P(X \geq 2) = 1 - P(X < 2)[/tex]
In which
[tex]P(X < 2) = P(X = 0) + P(X = 1)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{10,0}.(0.25)^{0}.(0.75)^{10} = 0.0563[/tex]
[tex]P(X = 1) = C_{10,1}.(0.25)^{1}.(0.75)^{9} = 0.1877[/tex]
[tex]P(X < 2) = P(X = 0) + P(X = 1) = 0.0563 + 0.1877 = 0.2440[/tex]
[tex]P(X < 2) = P(X = 0) + P(X = 1) = 1 - 0.244 = 0.756[/tex]
75.60% probability that the driller drills at 10 locations and has at least 2 success
