the venn diagram below shows the 9 students in mr. perezs class.
the diagram shows the memberships for the softball club and the music club.
note that \ivanna\ is outside the circles since she is not a member of either club.
one student from the class is randomly selected.
let $a$ denote the event \the student is in the softball club.\
let $b$ denote the event \the student is in the music club.\

(a) find the probabilities of the events below.
write each answer as a single fraction.
$p(a) = $
$p(b) = $
$p(a \text{ and } b) = $
$p(a \text{ or } b) = $
$p(a) + p(b) - p(a \text{ and } b) = $

(b) select the probability that is equal to $p(a) + p(b) - p(a \text{ and } b)$.
$p(a)$
$p(b)$
$p(a \text{ and } b)$
$p(a \text{ or } b)$

Question

the venn diagram below shows the 9 students in mr. perezs class.
the diagram shows the memberships for the softball club and the music club.
note that \ivanna\ is outside the circles since she is not a member of either club.
one student from the class is randomly selected.
let $a$ denote the event \the student is in the softball club.\
let $b$ denote the event \the student is in the music club.\

(a) find the probabilities of the events below.
write each answer as a single fraction.
$p(a) = $
$p(b) = $
$p(a \text{ and } b) = $
$p(a \text{ or } b) = $
$p(a) + p(b) - p(a \text{ and } b) = $

(b) select the probability that is equal to $p(a) + p(b) - p(a \text{ and } b)$.
$p(a)$
$p(b)$
$p(a \text{ and } b)$
$p(a \text{ or } b)$

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Venn Diagram Probability", "Addition Rule of Probability" ], "new_concepts": [], "current_concepts": [ "Venn Diagram Probability", "Addition Rule of Probability" ] } ## Count outcomes from the Venn diagram $$ \begin{aligned} &\text{Total students } (N) = 9 \ &\text{Softball Club } (A) = \{\text{Kala, Leila, Trey, Felipe, Keisha}\} \implies n(A) = 5 \ &\text{Music Club } (B) = \{\text{Keisha, Abdul, Mary, Diane}\} \implies n(B) = 4 \ &A \cap B = \{\text{Keisha}\} \implies n(A \text{ and } B) = 1 \ &A \cup B = \{\text{Kala, Leila, Trey, Felipe, Keisha, Abdul, Mary, Diane}\} \implies n(A \text{ or } B) = 8 \end{aligned} $$ ## Calculate probabilities for part (a) $$ \begin{aligned} &P(A) = \frac{5}{9} \ &P(B) = \frac{4}{9} \ &P(A \text{ and } B) = \frac{1}{9} \ &P(A \text{ or } B) = \frac{8}{9} \ &P(A) + P(B) - P(A \text{ and } B) = \frac{5}{9} + \frac{4}{9} - \frac{1}{9} = \frac{8}{9} \end{aligned} $$ ## Identify the equivalent probability for part (b) $$ \begin{aligned} &P(A) + P(B) - P(A \text{ and } B) = \frac{8}{9} \ &P(A \text{ or } B) = \frac{8}{9} \ &\text{Therefore, } P(A) + P(B) - P(A \text{ and } B) = P(A \text{ or } B) \end{aligned} $$ ## Question (a) $P(A) =$ $\frac{5}{9}$ $P(B) =$ $\frac{4}{9}$ $P(A \text{ and } B) =$ $\frac{1}{9}$ $P(A \text{ or } B) =$ $\frac{8}{9}$ $P(A) + P(B) - P(A \text{ and } B) =$ $\frac{8}{9}$ ## Question (b) $P(A)$

$P(B)$

$P(A \text{ and } B)$

$P(A \text{ or } B)$

{ "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Statistics", "Addition Rule of Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Count outcomes from the Venn diagram

Calculate probabilities for part (a)

Identify the equivalent probability for part (b)

Answer:

Question (a)

Question (b)