Question

question 10
let event a = you buy a new umbrella on friday. which event is most likely to be independent of a?
a. you visited the school lost-and-found on thursday afternoon.
b. the weather forecast says it will rain on saturday.
c. it rained on thursday.
d. you had a math test on thursday.

question 11
you roll two number cubes.
let event a = the number rolled on the first cube is 6.
let event b = the sum of the numbers rolled is 11.
what does $p(b|a)$ represent?
a. the probability that the sum is not 11
b. the probability that the first cube does not show a 6
c. the probability that the first number cube shows a 6, given that the sum is 11
d. the probability that the sum is 11, given that the first number cube shows a 6

question 12
this venn diagram shows sports played by 10 students.
let event a = the student plays basketball.
let event b = the student plays soccer.
what is $p(a|b)$?
a. $\frac{1}{10}=0.10$
b. $\frac{1}{4}=0.25$
c. $\frac{1}{6}\approx0.17$
d. $\frac{1}{3}\approx0.33$

Explanation:

Question 10

Two events are independent if the occurrence of one does not affect the probability of the other. Buying an umbrella on Friday (Event A) has no logical connection to having a math test on Thursday; the other options relate to weather or lost-and-found which could influence umbrella purchases.

In probability notation, $P(B|A)$ represents the conditional probability of event B occurring, given that event A has already occurred. Here, Event A is "first cube is 6" and Event B is "sum is 11".

Step1: Define conditional probability

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

Step2: Count relevant students

Students in $B$ (play soccer): Ella, Mickey, Marcus (total 3). Students in $A \cap B$ (play both): Ella (total 1).

Step3: Calculate the probability

$P(A|B) = \frac{1}{3} \approx 0.33$

Answer:

D. You had a math test on Thursday.

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Question 11