Question

MUTUALLY EXCLUSIVE EVENTS AND THE ADDITION RULE

Determine whether the following pair of events are mutually exclusive.

1) A card is drawn from a deck.  

C={It is a King} D={It is a heart}.

2) Two dice are rolled.

G={The sum of dice is 8} H={One die shows a 6}

3) A family has three children.

K={First born is a boy} L={The family has children of both sexes}

Use the addition rule to find the following probabilities.

1) A die is rolled, and the events E and F are as follows:

E={An even number shows}   F={A number greater than 3 shows}

Find P(E or F)

Answer 1

Solution:

Part i) Determine whether the following pair of events are mutually exclusive.

1) A card is drawn from a deck.  

C={It is a King}

D={It is a heart}.

There are 4 king cards out of which one is heart.

thus we have:

Since this event is not empty, events C and D are not mutually exclusive.

2) Two dice are rolled.

G={The sum of dice is 8}

G = { (2,6) , (3,5) , ( 4.4) , ( 5,3), (6,2) }

H={One die shows a 6}

H = { (1,6) ,(2,6) ,(3,6) ,(4,6) ,(5,6) ,(6,6) , (6,1) ,(6,2) ,(6,3) ,(6,4) ,(6,5) }

Thus we have:

Since this event is not empty,  events G and H are not mutually exclusive.

3) A family has three children.

Sample Space:

S = { BBB , BBG , BGB , GBB , GGB, GBG , BGG , GGG }

K={First born is a boy}

K= { BBB , BBG , BGB , BGG }

L={The family has children of both sexes}

L = {BBG , BGB , GBB , GGB, GBG , BGG }

Thus we have:

Since this event is not empty,  events L and K are not mutually exclusive.

Part ii) Use the addition rule to find the following probabilities.

1) A die is rolled, and the events E and F are as follows:

E={An even number shows}   F={A number greater than 3 shows}

Find P(E or F)=.........?

P(E or F)= P(E) + P(F) - P( E and F)

Sample Space = S = { 1,2,3,4,5,6}

E={An even number shows}

E = { 2,4,6}

P(E) = 3/6  

F={A number greater than 3 shows}

F = {4,5,6}

P(F) = 3/6  

and

E and F = { 4,6 }

P( E and F) = 2/6

Thus

P(E or F)= P(E) + P(F) - P( E and F)

P(E or F)= 3/6 + 3/6 - 2/6

P(E or F)= ( 3+3-2) / 6

P(E or F)= 4/6

P(E or F)=2/3

P(E or F)= 0.6667