Three hundred and thirty two families in the DMV area about their vacation habit

Question

Three hundred and thirty two families in the DMV area about their vacation habits. The accompanying two-way table shows the number of families according to where they live ( rural, suburban, urban) and the lengh of their last vacation (1 to 7 days, 8 days or more)

(1) Are the two events RURAL, 8 days or more independent? Prove or Disprove mathematically.

(2) Find the probability that a family spent 1 to 7 days on vacation.

(3) Find the probability that a family is SUBURBAN and spent 1 to 7 days on vacation.

(4) Given that the family is SUBURBAN, find the probability that they spent 1 tp 7 days vacation.

(5) Are the two events URBAN, 1 to 7 days mutually exclusive? Prove or Disprove mathematically.

RURAL SUBURBAN URBAN 1 to 7days 90 57 52 8 days or more 74 38 21

Explanation / Answer

(1) Are the two events RURAL, 8 days or more independent? Prove or Disprove mathematically.

P ( rural ) = (90 + 74)/ 332 = 0.4939

P ( 8 days or more) = (38 + 21 + 74)/ 332 = 0.40

for being two events independent

P(A and B) = P(A) · P(B)

P ( rural and 8 days or more ) = 74/332 = 0.2229

and P ( rural) * P( 8 days or more) = 0.4939 * 0.40 = 0.19756

so both events are not independent.

(2) Find the probability that a family spent 1 to 7 days on vacation.

probability that a family spent 1 to 7 days on vacation = P(1 to 7 days) = ( 90+ 57 + 52)/ 332 = 0.60

(3) Find the probability that a family is SUBURBAN and spent 1 to 7 days on vacation.

Pr ( Suburban and spent 1 to 7 days) = 57/ 332 = 0.1717

(4) Given that the family is SUBURBAN, find the probability that they spent 1 tp 7 days vacation.

Pr ( spent 1 to 7 days of suburban) = 57/ ( 57 + 38) = 0.60

(5) Are the two events URBAN, 1 to 7 days mutually exclusive? Prove or Disprove mathematically.

Pr (Urban) = (52 + 21)/ 332 = 0.22

Pr ( 1 to 7 days) = ( 90 + 57 + 52)/ 332 = 0.60

For two mutually exclusive events A and B P(A B) = 0

so Pr ( Urban and 1 to 7 days )= 52/332 = 0.1567 0 so not mutually exclusive events.

When two events, A and B, are independent, the probability of both occurring is:

P(A and B) = P(A) · P(B)

P ( rural and 8 days or more ) = 74/332 = 0.2229

and P ( rural) * P( 8 days or more) = 0.4939 * 0.40 = 0.19756

so both events are not independent.

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