Question

1. Let H be the event of observing a head and T be the event of observing a tail. A balanced coin is tossed three times.

(a) List all possible outcomes.

(b) List all possible outcomes and find the probabilities of the following events.

      A = event exactly two heads are tossed

      B = event the first toss is a tail

      C = event the first toss is a head

      D = event all three tosses come up the same

(c) List all possible outcomes and find the probabilities of the following events. (10 marks: 2 for each)

1. not A

2. A & B

3. B & C

4. C or D

5. D|A

(d) Identify all possible pairs of events defined in part (b) that are mutually exclusive.

(e) Are the events A and D independent? Explain your answer by calculation.

Answer 1

Let H be the event of observing a head and T be the event of observing a tail. A balanced coin is tossed three times.

a) Number of possible outcomes = 2*2*2 = 8

outcome on 1st toss	outcome on 2nd toss	outcome on 3rd toss	outcome on 3 tosses
H	H	H	HHH( ALL HEADS)
H	H	T	HHT(TWO HEADS)
H	T	H	HTH(TWO HEADS)
T	H	H	THH(TWO HEADS)
T	T	H	TTH(ONE HEAD)
T	H	T	THT(ONE HEAD)
H	T	T	HTT(ONE HEAD)
T	T	T	TTT(ALL TAILS)

b)

possible outcome	possibe number of such outcomes	probability
exactly two heads (event A)	3(HHT,THH,HTH)	3/8
the first toss is a tail (event B)	4(THH,TTH,THT,TTT)	4/8 =1/2
the first toss is a head( event C)	4(HTT,HHT,HTH,HHH)	4/8=1/2
all same (event D)	2(TTT,HHH)	2/8 = 1/4

C)

event	possible outcomes	probability	remarks
not A	(HHH,HTT,THT,TTH,TTT)	5/8	0,1,3 heads
A&B	HHT,HTH	2/8	two heads and 1st toss is head
B&C	NOT POSSIBLE	0	FIRST TOSS IS TAIL AND HEAD BOTH
C or D	HHH,TTT,HHT,HTH,HTT,	5/8
D|A	HHH	1/3 because here we use P(D|A) = P(D&A)/P(A) = n(D&A) / n(A) = 1/3	all same given exactly two heads

d) mutually exclusive events : two events which cannot occur simultaneously

e.g. getting both head and tail on same toss are mutually exclusive as they cannot occur simultaneously.

possible combinations are A&B, A&C,A&D,B&C,B&D,C&D

Lets check which one of them are mutually exclusive.

i) A&B--NOT MUTUALLY EXCLUSIVE

A is exactly two heads and B is first toss being tail

since both can occur together as THH so they are not mutually exclusive.

ii) B&C--MUTUALLY EXCLUSIVE

C is the event of first toss being head and B is first toss being tail

since both cannot occur together as first toss can give only one result ether head or tail but not both so they are mutually exclusive.

iii) A&C --

A is exactly two heads AND C is the event of first toss being head

possible case is HHT,HTH

HENCE A&C ARE NOT MUTUALLY EXCLUSIVE EVENTS.

iv) A&D

A is exactly two heads AND D is all same event since, both cannot occur as for that to happen the third toss will also have to be a head which is not possible for event A

hence ,A and D are mutually exclusive.

v) B&D

B is first toss being tail and D is all same event

and possible outcome is TTT

HENCE, B and D are not mutually exclusive.

vi) C&D

C is the event of first toss being head and D is all same event

and possible event is HHH

SO, C&D are not mutally exclusive.

so, only A&D and B&C are mutually exclusive events.

e) for events A&D to be independent P(A|D) should be equal to P(A) or P(A and D) = P(A).P(D)

now, P(A & D) = 0 as they are mutually exclusive events.

but P(A) = 3/8 and P(D) = 2/8 therefore, P(A).P(D) = (3/8).(2/8) = 3/32

since, P(A and D) IS NOT EQUAL to P(A).P(D)

HENCE, A&D are not independent events but they are dependent events.