Question

There are 210210210 students in a twelfth grade high school class. 909090 of these students have at least one sister and 105105105 have at least one brother. Out of these, there are 454545 who have at least one sister and one brother. Let AAA be the event that a randomly selected student in the class has a sister and BBB be the event that the student has a brother. Based on this information, answer the following questions. What is P(A)P(A)P, left parenthesis, A, right parenthesis, the probability that the student has a sister? What is P(B)P(B)P, left parenthesis, B, right parenthesis, the probability that the student has a brother? What is P(A\text{ and }B)P(A and B)P, left parenthesis, A, start text, space, a, n, d, space, end text, B, right parenthesis, the probability that the student has a sister and a brother? What is P(B\text{ | }A)P(B | A)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, the conditional probability that the student has a brother given that he or she has a sister? Is P(B\text{ | }A)=P(B)P(B | A)=P(B)P, left parenthesis, B, start text, space, vertical bar, space, end text, A, right parenthesis, equals, P, left parenthesis, B, right parenthesis? Are the events AAA and BBB independent?

Answer 1

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.

$P(E)=\frac{n(E)}{N}$

The conditional probability of an event X provided that another event Y has already occurred is:

$P(X|Y)=\frac{P(A\cap Y)}{P(Y)}$

If the events X and Y are independent then,

$P(X|Y)=P(X)$

(1)

Compute the probability of event A as follows:

$P(A)=\frac{n(A)}{N}\\\\=\frac{90}{210}\\\\=0.4286$

The value of P (A) is 0.4286.

(2)

Compute the probability of event B as follows:

$P(B)=\frac{n(B)}{N}\\\\=\frac{105}{210}\\\\=0.50$

The value of P (B) is 0.50.

(3)

Compute the probability of event A and B as follows:

$P(A\cap B)=\frac{n(A\cap B)}{N}\\\\=\frac{45}{210}\\\\=0.2143$

The value of P (A ∩ B) is 0.2143.

(4)

Compute the probability of B given A as follows:

$P(B|A)=\frac{P(A\cap B)}{P(A)}\\\\=\frac{0.2143}{0.4286}\\\\=0.50$

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.