Respuesta :
Answer:
[tex]P(X=0)=(15C0)(0.5)^0 (1-0.5)^{15-0}=0.0000305[/tex]
[tex]P(X=1)=(15C1)(0.5)^1 (1-0.5)^{15-1}=0.000457[/tex]
[tex]P(X=2)=(15C1)(0.5)^2 (1-0.5)^{15-2}=0.00320[/tex]
And adding the results we got:
[tex] P(X<3) =P(X \leq 2) = 0.0036875[/tex]
Step-by-step explanation:
We can define the variable of interest s X representing the number of correct questions for the exam. and we can model this random variable with a binomial distribution. The probability of select the correct answer would be [tex]p =\frac{1}{2}[/tex] since is a true/false question.
[tex] X \sim Binom (n =15, p=0.5[/tex]
And we want to find this probability:
[tex]P(X <3)= P(X\leq 2)=P(X=0) +P(X=1) +P(X=2)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
And we want to find this probability:
[tex]P(X \leq 2)=P(X=0)+P(X=1)+P(X=2)[/tex]
We can find the individual probabilities and we got:
[tex]P(X=0)=(15C0)(0.5)^0 (1-0.5)^{15-0}=0.0000305[/tex]
[tex]P(X=1)=(15C1)(0.5)^1 (1-0.5)^{15-1}=0.000457[/tex]
[tex]P(X=2)=(15C1)(0.5)^2 (1-0.5)^{15-2}=0.00320[/tex]
And adding the results we got:
[tex] P(X<3) =P(X \leq 2) = 0.0036875[/tex]
