In 2002, 33.9% of all fires were structure fires. Of these, 78% were residential

Question

In 2002, 33.9% of all fires were structure fires. Of these, 78% were residential fires. The causes of structure fire and the numbers of fires during 2002 for each cause are displayed in the next table. Suppose that four independent structure fires are reported in 1 day, and let X denote the number, out of the four, that are caused by cookingof the four, that are caused by cooking. Cause of Fire Number of Fires

Cooking 29,706

Chimney Fires 8,638

Incinerator 284

Fuel Burner 3,226

Commercial Compactor 246

Trash/Rubbish 9,906

Total 52,006

a Find the probability distribution for X in tabular form.

b Find the probability that at least one of the four fires was caused by cooking.

Explanation / Answer

a)

Here, the probability that a fire is caused by cooking is

P(cooking) = 0.339*0.78*(29706/52006) = 0.151037583

Note that the probability of x successes out of n trials is  
  
P(n, x) = nCx p^x (1 - p)^(n - x)  
  
where  
  
n = number of trials =    4
p = the probability of a success = 0.151037583
x = the number of successes

Hence, the table:

[ANSWER]

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b)

P(at least 1) = 1 - P(0) = 1 - 0.519462091 = 0.480537909 [ANSWER]

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