Respuesta :
Answer:
Step-by-step explanation:
The complete is question is as follows
Suppose that X is the number of successes in an experiment with 9 independent trials where the probability of success is 2/5 . Find each of the following probabilities. Round answers to the nearest ten-thousandth.
P (X < 2)
P(X ≥ 2)
Recall that given an event A, we have the following property [tex]P(A^c)=1-P(A)[/tex]. Note that if we consider the event A to be "X<2", its complement is the event "X≥ 2" then, the second probability is
P(X≥ 2) = 1- P(X<2). Therefore, we only need to calculate the first probability to solve the problem.
REcall that given a number of [tex]n[/tex] indepent trials of an experiment whose outcomes are "success" or "fail", where the probability of having a succes is p, the number of success is a random variable that is distributed as a binomial random variable. Therefore, we have that if X is the number of successes,
[tex]P(X=x) = \binom{n}{x} p^x(1-p)^{n-x}[/tex].
In our case, we have that p=2/5 and n=9. Hence,
[tex]P(X<2) = P(X=0) + P(X=1) = \binom{9}{0}(2/5)^0(3/5)^{9-0}+ \binom{9}{0}(2/5)^1(3/5)^{9-1} = 0.071[/tex]
Then, P(X≥ 2) = 1-0.071 = 0.929
The complete question is:
Suppose that X is the number of successes in an experiment with 9 independent trials where the probability of success is 2/5 . Find each of the following probabilities. Round answers to the nearest ten-thousandth.
(i) P (X < 2)
(ii) P(X ≥ 2)
Answer:
(i) P(x < 2) = 0.07054
(ii) P(x ≥ 2) = 0.92946
Step-by-step explanation:
Given that X is the number of successes in an experiment n = 9, and the probability of success p = 2/5
If x is approximated Binomial(n, p)
Then
P(x) = nCxP^xq^(n - x)
Where q = 1 - p
Here, q = 1 - (2/5) = 3/5
And nCx, read as "n combination x"
is given as n!/(n - x)! x!
P(x < a) = P(x = a - 1) + P(x = a - 2) + ... + P(x = 1) + P(x = 0)
And
P(x ≥ a) = 1 - P(x < a)
Now, P(x < 2)
= P(x = 1) + P(x = 0)
= [9C0(2/5)^0 (3/5)^(9 - 0)] + [9C1(2/5)^1 (3/5)^(9 - 1)]
= [1×1×(3/5)^9] + [9 × (2/5) × (3/5)^8]
= 0.070543872
≈ 0.07054 (To the nearest ten thousandth)
Now,
P(x ≥ 2) = 1 - P(x < 2)
= 1 - 0.07054
= 0.92946
