Question

Please answer this question immediatly

Answer 1

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

Answer 2

`\sf a) \quad P(contains\:apple)+P(contains\:blueberry)=(5)/(6)`

`\sf b) \quad P(contains\:apple\:or\:blueberry)=(11)/(15)`

c) Not mutually exclusive events

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 = (Number\:of\:ways\:it\:can\:occur)/(Total\:number\:of\:possible\:outcomes)`

Let A = contains apple

Let B = contains blueberry

`\implies\sf P(A)=(15)/(30)=(1)/(2)`

`\implies\sf P(B)=(10)/(30)=(1)/(3)`

Part (a)

`\begin{aligned} \implies \sf P(A)+P(B) & =\sf (1)/(2)+(1)/(3)\\\\ & = \sf (3)/(6)+(2)/(6)\\\\ & = \sf (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)=(5)/(6) \quad \textsf{(from part a)}`

`\sf P(A\:and\:B)=(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 (5)/(6)-(3)/(30)\\\\ & =\sf (25)/(30)-(3)/(30)\\\\ & =\sf (22)/(30)\\\\ & = \sf (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) = (12+3+7)/(30)=(22)/(30)=(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)=(5)/(6) \quad \textsf{(from part a)}`

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

`\sf As\:(11)/(15)\\eq (5)/(6) \implies P(A \: or \: B)\\eq P(A)+P(B)`

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