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QuestionA postal carrier delivered letters, ads, and magazines. Find probabilities for various scenarios based on the total distributions.

Studdy Solution

STEP 1

Assumptions1. The total number of items distributed is879. . The number of items delivered to homes is434.
3. The number of items delivered to businesses is445.
4. The total number of letters distributed is185.
5. The total number of ads distributed is546.
6. The total number of magazines distributed is148.
7. The number of letters delivered to homes is125.
8. The number of ads delivered to homes is206.
9. The number of magazines delivered to homes is103.
10. The number of letters delivered to businesses is60.
11. The number of ads delivered to businesses is340.
12. The number of magazines delivered to businesses is45.

STEP 2

We will calculate the probability of each event by dividing the number of successful outcomes by the total number of outcomes.
a) The probability that an item went to a home is the number of items delivered to homes divided by the total number of items.
(Home)=Number of items delivered to homesTotal number of items(\text{Home}) = \frac{\text{Number of items delivered to homes}}{\text{Total number of items}}

STEP 3

Substitute the given values into the equation.
(Home)=434879(\text{Home}) = \frac{434}{879}

STEP 4

Calculate the probability.
(Home)=0.494(\text{Home}) =0.494

STEP 5

b) The probability that an item was a magazine OR a letter is the sum of the probabilities of each event.(Magazine or Letter)=(Magazine)+(Letter)(\text{Magazine or Letter}) =(\text{Magazine}) +(\text{Letter})

STEP 6

Substitute the given values into the equation.
(Magazine or Letter)=148879+185879(\text{Magazine or Letter}) = \frac{148}{879} + \frac{185}{879}

STEP 7

Calculate the probability.
(Magazine or Letter)=0.379(\text{Magazine or Letter}) =0.379

STEP 8

c) The probability that an item was a magazine AND went to a home is the number of magazines delivered to homes divided by the total number of items.
(Magazine and Home)=Number of magazines delivered to homesTotal number of items(\text{Magazine and Home}) = \frac{\text{Number of magazines delivered to homes}}{\text{Total number of items}}

STEP 9

Substitute the given values into the equation.
(Magazine and Home)=103879(\text{Magazine and Home}) = \frac{103}{879}

STEP 10

Calculate the probability.
(Magazine and Home)=0.117(\text{Magazine and Home}) =0.117

STEP 11

d) The probability that an item was a magazine OR went to a home is the sum of the probabilities of each event minus the probability of both events occurring (since we are double counting this case).
(Magazine or Home)=(Magazine)+(Home)−(Magazine and Home)(\text{Magazine or Home}) =(\text{Magazine}) +(\text{Home}) -(\text{Magazine and Home})

STEP 12

Substitute the given values into the equation.
(Magazine or Home)=148879+434879−103879(\text{Magazine or Home}) = \frac{148}{879} + \frac{434}{879} - \frac{103}{879}

STEP 13

Calculate the probability.
(Magazine or Home)=0.544(\text{Magazine or Home}) =0.544

STEP 14

e) The probability that an item was a letter GIVEN that it went home is the number of letters delivered to homes divided by the number of items delivered to homes.
(Letter | Home)=Number of letters delivered to homesNumber of items delivered to homes(\text{Letter | Home}) = \frac{\text{Number of letters delivered to homes}}{\text{Number of items delivered to homes}}

STEP 15

Substitute the given values into the equation.
(Letter | Home)=125434(\text{Letter | Home}) = \frac{125}{434}

STEP 16

Calculate the probability.
(Letter | Home)=0.288(\text{Letter | Home}) =0.288

STEP 17

f) If two items are selected randomly with replacement, the probability of selecting two letters is the probability of selecting a letter multiplied by itself (since the events are independent).
(Two letters)=(Letter)2(\text{Two letters}) =(\text{Letter})^2

STEP 18

Substitute the given values into the equation.
(Two letters)=(185879)2(\text{Two letters}) = \left(\frac{185}{879}\right)^2

STEP 19

Calculate the probability.
(Two letters)=.044(\text{Two letters}) =.044The probabilities for each event are as followsa) The item went to a home.494b) The item was a magazine OR a letter.379c) The item was a magazine AND went to home.117d) The item was a magazine OR went home.544e) The item was letter GIVEN that it went home.288f) If two items are selected randomly with replacement, the probability of selecting two letters is.044.

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