Question

Math

Posted 5 months ago

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There are 300 students in a twelfth grade high school class. 120 of these students have at least one sister and 100 have at least one brother. Out of the students that have at least one brother, there are 40 who have at least one sister and one brother.
Let $A$ be the event that a randomly selected student in the class has a sister and $B$ be the event that the student has a brother. Based on this information, answer the following questions.
What is $P(A)$, the probability that the student has a sister?
What is $P(B)$, the probability that the student has a brother?
What is $P(A$ and $B)$, the probability that the student has a sister and a brother?
What is $P(A \mid B)$, the conditional probability that the student has a sister given that he or she has a brother?
Is $P(A \mid B)=P(A)$ ? Are the events $A$ and $B$ independent?
Choose all answers that apply:
A Yes, $P(A \mid B)=P(A)$.
B No, $P(A \mid B) \neq P(A)$.
c Yes, events $A$ and $B$ are independent events.
D No, events $A$ and $B$ are not independent events.
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Answer from Sia

Posted 5 months ago

Solution by Steps

step 2

To find $P(B)$, the probability that a student has a brother, we use the number of students with at least one brother (100) and divide by the total number of students (300). Using the asksia-ll calculation: $100/300 = 1/3 \approx 0.3333333333 = 33.33\%$

step 3

To find $P(A \text{ and } B)$, the probability that a student has both a sister and a brother, we use the number of students with at least one sister and one brother (40) and divide by the total number of students (300). Using the asksia-ll calculation: $40/300 = 2/15 \approx 0.1333333333 = 13.33\%$

step 4

To find $P(A | B)$, the conditional probability that a student has a sister given that they have a brother, we use the number of students with both a sister and a brother (40) and divide by the number of students with at least one brother (100). Using the asksia-ll calculation: $40/100 = 2/5 = 0.4 = 40\%$

step 5

To determine if $P(A | B) = P(A)$, we compare the probabilities from step 1 and step 4. Since both are $0.4$, we have $P(A | B) = P(A)$

step 6

To determine if events A and B are independent, we check if $P(A \text{ and } B) = P(A) \times P(B)$. Using the asksia-ll calculation: $P(A) \times P(B) = 0.4 \times 0.3333333333 \approx 0.1333333333$, which is equal to $P(A \text{ and } B)$. Therefore, events A and B are independent

[1] Answer

A

[2] Answer

C

Key Concept

Independence of Events

Explanation

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other, which can be mathematically expressed as $P(A \text{ and } B) = P(A) \times P(B)$.

Key Concept

Conditional Probability

Explanation

The conditional probability $P(A | B)$ is the probability of event A occurring given that event B has occurred, and is calculated as $P(A | B) = P(A \text{ and } B) / P(B)$. If $P(A | B) = P(A)$, then events A and

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