Respuesta :
Answer:
(1) The value of P (A) is 0.4286.
(2) The value of P (B) is 0.50.
(3) The value of P (A ∩ B) is 0.2143.
(4) The the value of P (B|A) is 0.50.
(5) The events A and B are independent.
Step-by-step explanation:
The events are defined as follows:
A = a student in the class has a sister
B = a student has a brother
The information provided is:
N = 210
n (A) = 90
n (B) = 105
n (A ∩ B) = 45
The probability of an event E is the ratio of the favorable number of outcomes to the total number of outcomes.
[tex]P(E)=\frac{n(E)}{N}[/tex]
The conditional probability of an event X provided that another event Y has already occurred is:
[tex]P(X|Y)=\frac{P(A\cap Y)}{P(Y)}[/tex]
If the events X and Y are independent then,
[tex]P(X|Y)=P(X)[/tex]
(1)
Compute the probability of event A as follows:
[tex]P(A)=\frac{n(A)}{N}\\\\=\frac{90}{210}\\\\=0.4286[/tex]
The value of P (A) is 0.4286.
(2)
Compute the probability of event B as follows:
[tex]P(B)=\frac{n(B)}{N}\\\\=\frac{105}{210}\\\\=0.50[/tex]
The value of P (B) is 0.50.
(3)
Compute the probability of event A and B as follows:
[tex]P(A\cap B)=\frac{n(A\cap B)}{N}\\\\=\frac{45}{210}\\\\=0.2143[/tex]
The value of P (A ∩ B) is 0.2143.
(4)
Compute the probability of B given A as follows:
[tex]P(B|A)=\frac{P(A\cap B)}{P(A)}\\\\=\frac{0.2143}{0.4286}\\\\=0.50[/tex]
The the value of P (B|A) is 0.50.
(5)
The value of P (B|A) = 0.50 = P (B).
Thus, the events A and B are independent.
