Question

Please answer this question immediatly

Answer 1

Answer:

$\sf a) \quad P(contains\:apple)+P(contains\:blueberry)=\dfrac{5}{6}$

$\sf b) \quad P(contains\:apple\:or\:blueberry)=\dfrac{11}{15}$

c)    Not mutually exclusive events

Step-by-step explanation:

From inspection of the Venn diagram:

• Total number of smoothies = 12 + 3 + 7 + 8 = 30
• Number of smoothies containing Apple = 12 + 3 = 15
• Number of smoothies containing Blueberry  = 3 + 7 = 10
• Number of smoothies containing both Apple and Blueberry = 3
• Number of smoothies not containing Apple or Blueberry = 8

$\sf Probability\:of\:an\:event\:occurring = \dfrac{Number\:of\:ways\:it\:can\:occur}{Total\:number\:of\:possible\:outcomes}$

Let A = contains apple

Let B = contains blueberry

$\implies\sf P(A)=\dfrac{15}{30}=\dfrac{1}{2}$

$\implies\sf P(B)=\dfrac{10}{30}=\dfrac{1}{3}$

Part (a)

$\begin{aligned} \implies \sf P(A)+P(B) & =\sf \dfrac{1}{2}+\dfrac{1}{3}\\\\ & = \sf \dfrac{3}{6}+\dfrac{2}{6}\\\\ & = \sf \dfrac{5}{6}\end{aligned}$

Part (b)

$\textsf{Addition Law}: \quad\sf P(A\:or\: B) = P(A)+P(B)-P(A\:and\:B)$

Given:

$\sf P(A)+P(B)=\dfrac{5}{6} \quad \textsf{(from part a)}$

$\sf P(A\:and\:B)=\dfrac{3}{30}\quad \textsf{(where the circles overlap)}$

$\begin{aligned}\implies \sf \sf P(A\:or\: B) &= \sf P(A)+P(B)-P(A\:and\:B)\\\\& =\sf \dfrac{5}{6}-\dfrac{3}{30}\\\\ & =\sf \dfrac{25}{30}-\dfrac{3}{30}\\\\ & =\sf \dfrac{22}{30}\\\\ & = \sf \dfrac{11}{15}\end{aligned}$

Or, we can simply read P(contains apple or blueberry) from the Venn diagram.  

P(A or B) is the total of the numbers inside the circles divided by the total number of smoothies:

$\sf P(A\:or\:B) = \dfrac{12+3+7}{30}=\dfrac{22}{30}=\dfrac{11}{15}$

Part (c)

For two events, A and B, where A and B are mutually exclusive:

$\sf P(A \: or \: B)=P(A)+P(B)$

Given:

$\sf P(A)+P(B)=\dfrac{5}{6} \quad \textsf{(from part a)}$

$\sf P(A\:or\:B)=\dfrac{11}{15} \quad \textsf{(from part b)}$

$\sf As\:\dfrac{11}{15}\neq \dfrac{5}{6} \implies P(A \: or \: B)\neq P(A)+P(B)$

Therefore, choosing a smoothie containing apple and choosing a smoothie containing blueberry are NOT mutually exclusive events.

Answer 2

Apple be A and blueberries be B

So

As per Venn diagram

• n(A)=12
• n(B)=7
• n(A$\cap$B)=3

#1

• n(A)+n(B)
• 12+7
• 19

P(A)=12/24

P(B)=7/24

Add

• 12+7/24
• 19/24

#2.

n(AUB)

• n(A)+n(B)-n(An B).
• 19-3
• 16

E=8+16=24

P(AUB)=16/24=2/3