Question

In: Math

In a town, 36% of the citizens contributed to the Republicans, 46% contributed to the Democrats,...

In a town, 36% of the citizens contributed to the Republicans, 46% contributed to the Democrats, and 12% contributed to both. What percentage contributed to neither party?

A box contains 4 white, 3 red, and 3 black marbles. One marble is chosen at random, and it is not black. Find the probability that it is white. (Enter your answer as a fraction.)

Suppose that 90% of drivers are "careful" and 10% are "reckless." Suppose further that a careful driver has a 0.2 probability of being in an accident in a given year, while for a reckless driver the probability is 0.3. What is the probability that a randomly selected driver will have an accident within a year? (Enter your answer to two decimal places.)

Solutions

Expert Solution

1 . In a Town :

  • 36 % citizens contributed to Republicans ( R ) i.e. , P ( R ) = 0.36
  • 46 % citizens contributed to Democrats ( D ) i.e. , P ( D ) = 0.46
  • 12 % citizens contributed to both i.e. , P ( R D ) = 0.12

Percentage citizens who contributed to neither party = P ( R ' D ' )

By Axiom of Probability , we know that :

  • P ( A B ) = P ( A ) + P ( B ) - P ( A B )
  • P ( A' B' ) = P ( A B ) ' = 1 - P ( A ​​​​​​​ B )

P ( R D ) = P ( R ) + P ( D ) - P ( R D ) = 0.36 + 0.46 - 0.12 = 0.70

  P ( R ' D ' ) =  P ( R D ) ' = 1 - P ( R ​​​​​​​ D ) = 1 - 0.70 = 0.30 = 30 %

2 . A Box Contains 10 Marbles of which :

  • 4 are White ( W ) Marbles
  • 3 are Red ( R ) Marbles
  • 3 are Black ( B ) marbles

A Marble is chosen at random and it is not black . Thus , our Sample space cuts down to 7 Marbles of which 4 are White and 3 are Red .

P ( Chosen Marble is White ) = 4 / 7

3 . Given that :

  • 90 % of Drivers are Careful ( C ) i.e. , P ( C ) = 0.9
  • 10 % of Drivers are Reckless ( R ) i.e. , P ( R ) = 0.1
  • Careful Driver ( C ) has 20 % chance of having Accident ( A ) i.e. , P ( A | C ) = 0.2
  • Reckless Driver ( R ) has 30 % chance of having Accident ( A ) i.e. , P ( A | R ) = 0.3

We use Law of Total Probability which states that :

P ( X ) =   P ( X | Y i ) * P ( Y i )

P ( Randomly Selected Driver will have Accident within a year )

P ( A ) = [ P ( A | C ) * P ( C ) ] + [ P ( A | R ) * P ( R ) ]

= 0.2 ( 0.9 ) + 0.3 ( 0.1 ) = 0.18 + 0.03 = 0.21


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