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4.  Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

(ii)  simple?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!] S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT} Now,  A = Event that three heads show up = {HHH}  B = Event that two heads and one tail show up = {HHT, HTH, THH}  C = Event that three tails show up = {TTT} D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT} (ii).If an event X has only one...

4.  Three coins are tossed once. Let A denote the event ‘three heads show”, B denote the event “two heads and one tail show”, C denote the event” three tails show and D denote the event ‘a head shows on the first coin”. Which events are

(i)  mutually exclusive?

Sample space when three coins are tossed = [Sample space when a coin is tossed thrice!] S = {HHH, HHT, HTH, HTT, THH, TTH, THT, TTT} Now,  A = Event that three heads show up = {HHH}  B = Event that two heads and one tail show up = {HHT, HTH, THH}  C = Event that three tails show up = {TTT} D = Event that a head shows on the first coin = {HHH, HHT, HTH, HTT} (i).  For two elements X and Y to be...

3.     An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

Which pairs of these events are mutually exclusive?

For two elements to be mutually exclusive, there should not be any common element amongst them. Also, A = {(3,6), (4,5), (5, 4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)} B = {(1,2), (2,2), (3, 2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)} C = {(3,6), (6,3), (5, 4), (4,5), (6,6)}  Now,  A  B =       (no common element in A and B) Hence, A and B are mutually...

3.     An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

C: the sum is at least $7$ and a multiple of $3$

Sample space when a die is rolled: S = {1, 2, 3, 4, 5, 6} Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes] E = [ {(x,y): x,y  S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)} Now, C: the sum is at least 7 and a multiple of 3 The sum can only be 9 or 12. C = [ {(a,b): (a,b)  E, a+b>6 & a+b = 3k, k  I} ]= {(3,6), (6,3), (5, 4), (4,5), (6,6)}

3.    An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

B: $2$ occurs on either die

Sample space when a die is rolled: S = {1, 2, 3, 4, 5, 6} Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes] E = [ {(x,y): x,y  S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)} Now, B: 2 occurs on either die Hence the number 2 can come on first die or second die or on both the die simultaneously. B = [ {(a,b): (a,b)  E, a or b = 2 } ]=...

3.    An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:

A: the sum is greater than $8$

Sample space when a die is rolled: S = {1, 2, 3, 4, 5, 6} Let E = Event of rolling a pair of dice (= Event that a die is rolled twice!) [6x6 = 36 possible outcomes] E = [ {(x,y): x,y  S } ] = {(1,1), (1,2)...(1,6),(2,1).....(6,5),(6,6)} Now, A : the sum is greater than 8 Possible sum greater than 8 are 9, 10, 11 and 12 A = [ {(a,b): (a,b)  E, a+b>8 } ]= {(3,6), (4,5), (5, 4), (6,3), (4,6),...

2.(vi)    A die is thrown. Describe the following events:

Also find (i)  ${F}'$

F = {3, 4, 5, 6}  F' = {3, 4, 5, 6}' = S - F = {1, 2}

2.(vi)     A die is thrown. Describe the following events:

Also find (h) $E\cap F'$

E = {6} F = {3, 4, 5, 6}  F' = {3, 4, 5, 6}' = S - F = {1, 2}  E  F' = {6}  {1, 2} =

2.(vi)     A die is thrown. Describe the following events:

Also find (g)  $D-E$

D = {1, 2, 3} E = {6}  D - E = {1, 2, 3} - {6}  = {1, 2, 3}

2.(vi)         A die is thrown. Describe the following events:

Also find  (f)  $A-C$

A = {1, 2, 3, 4, 5, 6} C = {3, 6}  A - C = {1, 2, 3, 4, 5, 6} - {3, 6}  = {1, 2, 4, 5}

2.(vi)      A die is thrown. Describe the following events:

Also find (e) $D\cap E$

D = {1, 2, 3} E = {6}  D  E = {1, 2, 3}   {6} =  (As nothing is common in these sets)

2.(vi)     A die is thrown. Describe the following events:

(d)    Also find  $E\cap F$

E = {6} F = {3, 4, 5, 6}  E  F = {6}   {3, 4, 5, 6} = {6}

2.(vi)     A die is thrown. Describe the following events:

Also find (c)  $B\cup C$

B= C= {3, 6}  B  C =   {3, 6} = {3, 6}

2.(vi)        A die is thrown. Describe the following events:

Also find (b) $A\cap B$.

A = {1, 2, 3, 4, 5, 6} B=  A  B = {1, 2, 3, 4, 5, 6}   =

2.(vi)  A die is thrown. Describe the following events:

Also find  (a)  $A\cup B$

A = {1, 2, 3, 4, 5, 6} B=  A  B = {1, 2, 3, 4, 5, 6}   = {1, 2, 3, 4, 5, 6}

2.   A die is thrown. Describe the following events:

(vi)  F: a number not less than 3

When a die is rolled, the sample space of possible outcomes: S = {1, 2, 3, 4, 5, 6} or {x : x  N, x<7} Given, F : a number not less than 3  F = {x: x  S, x  3 } = {3, 4, 5, 6}

2.  A die is thrown. Describe the following events:

(v)   E: an even multiple greater than 4

When a die is rolled, the sample space of possible outcomes: S = {1, 2, 3, 4, 5, 6} or {x : x  N, x<7} Given, E : an even number greater than 4 S1 = Subset of S containing even numbers = {2,4,6} Therefore , E = {6}

2.   A die is thrown. Describe the following events:

(iv)   D: a number less than 4

When a die is rolled, the sample space of possible outcomes: S = {1, 2, 3, 4, 5, 6} or {x : x  N, x<7} Given, D : a number less than 4 D = {1, 2, 3}

2.    A die is thrown. Describe the following events:

(iii) C: a multiple of 3.

When a die is rolled, the sample space of possible outcomes: S = {1, 2, 3, 4, 5, 6} or {x : x  N, x<7} Given, C : a multiple of 3 C = {3, 6}

2.     A die is thrown. Describe the following events:

(ii) B: a number greater than 7

When a die is rolled, the sample space of possible outcomes: S = {1, 2, 3, 4, 5, 6} or {x : x  N, x<7} Given, B: a number greater than 7 As no number on the die is greater than 7 B =
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