Question

A fair coin is tossed three times and the events AA, BB, and CC are defined as follows:

A:{A:{ At least one head is observed }} 
B:{B:{ At least two heads are observed }} 
C:{C:{ The number of heads observed is odd }}

Find the following probabilities by summing the probabilities of the appropriate sample points (note that 0 is an even number):

(a)  P(not C)P(not C) ==  
(b)  P((not A) and B)P((not A) and B) ==  
(c)  P((not A) or B or C)P((not A) or B or C) == 

Answer 1

A fair coin is tossed three times then possible outcomes are ,

{ HHH , HHT , HTH , THH , TTH , THT , HTT , TTT }

Total outcomes = 8

a)

We have to find probability of not c , i.e C complement ( C^c)

Event C:{ The number of heads observed is odd } = { 1 or 3 heads observed }

C = { TTH , THT , HTT, HHH }

P( C ) = 4/8        { Probability = Number of favorable outcome / Total outcome.}

P( Not C ) = 1 - P( C ) = 1 - 4/8 = 4/8 = 1/2

P( Not C ) = 1/2

b)

We have to find P((not A) and B)

Event A: { At least one head is observed }

A = { TTH , THT, HTT, HHT, HTH , THH ,HHH }

Not A = A^c = { TTT }

Event B:{ At least two heads are observed }

B = { HHT , HTH , THH , HHH}

( Not A ) and B = Empty set =

So, P(( Not A ) and B) = 0

c)

We have to find P((not A) or B or C)

Event A: { At least one head is observed }

A = { TTH , THT , HTT , HHT , HTH , THH ,HHH }

Not A = { TTT }

Event B:{ At least two heads are observed }

B = { HHT , HTH , THH , HHH}

Event C:{ The number of heads observed is odd }

C = {TTH , THT , HTT, HHH }

(not A) or B or C = { TTT , HHT , HTH , THH , HHH, TTH , THT, HTT }

#((not A) or B or C) = 8

P((not A) or B or C) = 8/8 = 1