Question

quiz 12.3.3: conditional probabilityquestion 1which situation involves a conditional probability?a. the probability that you win the first round of a gameb. the probability that you are given a skateboard for your birthdayc. the probability that you roll a 3 on a number cubed. the probability that you win the second round of a game given that you won the first roundquestion 2you throw two darts at a dartboard.let event a = the first dart hits the center.let event b = the second dart hits the center.what does $p(b|a)$ represent?a. the probability that the second dart doesnt hit the centerb. the probability that either the first dart or the second dart hits the centerc. the probability that the second dart hits the center given that the first dart hits the centerd. the probability that the first dart hits the center given that the second dart hits the centerquestion 3a group of 10 students participate in chess club, karate club, or neither.let event a = the student is in chess club.let event b = the student is in karate club.one of these students is randomly selected. what is $p(a|b)$?a. $\frac{2}{6} approx 0.33$b. $\frac{2}{10} = 0.20$c. $\frac{6}{10} = 0.60$d. $\frac{4}{6} approx 0.67$

Explanation:

Question 1

Step1: Define conditional probability

Conditional probability is the probability of an event occurring given that another event has already occurred.

Step2: Analyze each option

• Option A: No prior condition, just a single event probability.
• Option B: No prior condition, just a single event probability.
• Option C: No prior condition, just a single event probability.
• Option D: Probability of winning the second round given winning the first, which fits conditional probability.

Question 2

Step1: Recall conditional probability notation

$P(B|A)$ denotes the probability of event $B$ occurring given that event $A$ has already occurred.

Step2: Match to given events

Event $A$ = first dart hits center, Event $B$ = second dart hits center. So $P(B|A)$ is the probability the second dart hits center given the first did.

Question 3

Step1: Define $P(A|B)$ formula

$P(A|B) = \frac{\text{Number of students in both } A \text{ and } B}{\text{Number of students in } B}$

Step2: Count relevant students

• Students in both $A$ (chess) and $B$ (karate): Yolanda, Rob → 2 students
• Students in $B$ (karate): Yolanda, Rob, Viki, Sam, Tuan, Uyen → 6 students

Step3: Calculate the probability

$P(A|B) = \frac{2}{6} \approx 0.33$

Answer:

1. D. The probability that you win the second round of a game given that you won the first round
2. C. The probability that the second dart hits the center given that the first dart hits the center
3. A. $\frac{2}{6} \approx 0.33$