Probability 1. Let A, B and C be three events in a sample space S. Express the f

Question

Probability

1. Let A, B and C be three events in a sample space S. Express the following events in term of these sets: Only A occurs At least one of the three events occurs. At least two of the events occur At most one of the three events occurs.None of the events occurs. » A and B occur, but C does not. » Exactly two of the events occur Not more than two of the events occur. 2. Toss a coin labeled by Head and Tail n times. For 1 Sjínthe event Aj occurs if 1 P the jth toss is Head. Describe the following events B1,... , B1 in terms of the A,'s: Bi = {All tosses are Heady, B2 = {At least one toss is Head), Be = {Exactly one toss is Heady and {All tosses show the same side of the coin Determine the cardinality of B1,..., B

Explanation / Answer

1. Only A occurs

A - (A B) - (A C)

A and B occur but C does not

(A B) - C

Atleast one of the three events occur

A U B U C

Atleast two of the events occur

(A B) U (A C) U (B C)

Atmost one of the three events occur

[(A B) U (A C) U (B C)]c

None of the events occur

[A U B U C]c

Exactly two of the events occur

[(A B) U (A C) U (B C)] - (A B C)

Not more than two of the events occur

(A B C)c

2. B1 = {All tosses are head}

B1 = A1 A2 A3 ..... An

The cardinality of B1 is exactly 1 as there is only one event of all tosses returning heads.

B2 = {Atleast one toss is a head}

B2 = A1 U A2 U A3 ..... An

The cardinality of B2 is 2n as the number of heads is actually the number of subsets of A1, A2 .... An.

B3 = {Exactly one toss is a head}

B3 = (A1 - A2 - A3 ..... An) U (A2 - A1 - A3 ..... An) U (A3 - A1 - A2 ..... An) ....... (An - A1 - A2 ..... An-1)

The cardinality of B3 is n as the one toss with head could be any of the n values.

B4 = {All tosses show the same side of the coin}

B4 = (A1 A2 A3 ..... An) U (A1c A2c A3c ..... Anc)

which can also be written as (A1 A2 A3 ..... An) U (A1 U A2 U A3 ..... An)c by De-Morgan's law

The cardinality of B4 is 2 as there are only two possible events viz. all heads or all tails.

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