Tut 30 POYO (Solutions)

7/24/2019 Tut 30 POYO (Solutions)

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Tutorial 30: Probability II

Practise on Your Own

1 A, B and C fire one shot each at a target. The probability that A will hit the target is5

1. The

probability that B will hit the target is4

1

. The probability that C will hit the target is3

1

. If

they fire together, calculate the probability that

(i) all three shots hit the target,

(ii) only Cs shot hits the target,

(iii) at least one shot hits the target,

(iv) e!actly two shot hit the target.

2 A recent poll of residents in a certain country re"ealed the following infor#ation about "oters

preferences$

The following e"ents are defined$

M$ resident is #ale%F$ resident is fe#ale%D$ resident is a &e#ocrat%R$ resident is a 'epublicanI$ resident is an independent.

(ind$ )a* +)RF*, )b* +)MD*, )c* +)MRI *.

3 I"an and -oe are chess players. In any ga#e the probabilities of I"an beating -oe, -oe beating

I"an, or the ga#e resulting in a draw are ./, .1 or .3 respecti"ely. They play a series of 5

ga#es. Calculate the probability that

(i) they win alternate ga#es with I"an winning the first ga#e,

(ii) e!actly three of the ga#es are drawn,

(i) -oe does not win #ore than one ga#e.

Democrat Reublican In!een!ent

"ale 5 4 0

#emale / 3 05

1


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$ A bo! contains 05 apples, of which 0 are red and 5 are green. f the red apples, 3 contain

#aggots and of the green apples, 1 contains #aggots. Two apples are chosen at rando# fro#

the bo!. (ind, in any order, the probability that

(i) both apples contain #aggots,

(ii) both apples are red and at least one contains #aggots,

(iii) at least one apple contains #aggots, gi"en that both apples are red,

(iv) both apples are red gi"en that at least one apple is red. 2

% A choir has sopranos, / altos, 3 tenors and 4 basses. The sopranos and altos are wo#en and

the tenors and basses are #en. At a particular rehearsal, three #e#bers of the choir are chosen

at rando# to #ae tea. (ind the probability that

(i) all three tenors are chosen,

(ii) e!actly one bass is chosen,

(iii) two wo#en are chosen, gi"en that e!actly one bass is chosen,

(iv) the chosen group contains e!actly one tenor or e!actly one bass, or both.

0


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& In a si#ple #odel of the weather in ctober, each day is classified as either fine or rainy.

The probability that a fine day is followed by a fine day is ..

The probability that a rainy day is followed by a fine day is .4.

The probability that 1 ctober is fine is .5. (ind the probability that

(i) 0 ctober is fine

(ii) 3 ctober is fine.(iii) (ind the conditional probability that 3 ctober is rainy, gi"en that 1 ctober is fine.

(iv) (ind the conditional probability that 1 ctober is fine, gi"en that 3 ctober is rainy.

2-6707/8

' A bag contains 1 orange9fla"oured, 14 strawberry9fla"oured and 1/ cherry9fla"oured sweets

which are of identical shapes and si:es.

Benny selects a sweet at rando# fro# the bag. If it is not cherry9fla"oured he replaces it and

selects another sweet at rando#. ;e repeats the process until he obtains a cherry9fla"oured

sweet. Calculate the probability that

(i) the first sweet selected is strawberry9fla"oured and the fourth sweet is orange9fla"oured,

(ii) he selects an e"en nu#ber of sweets.

olution*

(i) 'e


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+ A street ball #atch between =ingapore >nited and =ingapore 'o"ers ends in a draw and the

#atch is to be decided on the result of penalty ics. ne round of penalty ics consists of two

ics, one ic being taen by each tea#. If both tea#s score or if both tea#s fail to score, then

another round of penalty ics is played. The #atch is won when, in a round of penalty ics,

one tea# scores and the other tea# fails to score.

The probability that =ingapore >nited scores a goal with a penalty ic is . and

independently, the probability that =ingapore 'o"ers scores a goal with a penalty ic is .6.(ind the probability that the result of the #atch is still undecided after 1 round.

(i) (ind the probability that =ingapore >nited wins the #atch in less than 3 rounds of penalty

ics gi"en that =ingapore >nited scores a goal in the first round of penalty ics.

(ii) (ind the least nu#ber of rounds of penalty ics nsuch that the probability that the result

of the #atch is decided in at #ost n rounds of penalty ics is greater than .6.

0,-.//-II-13*

olution*

+)#atch is still undecided after 1 round *

? +)both score a goal* @ +)both did not score a goal* ? . .6 @ .0 .1 ? .4 ?5.

3

(i) +)>nited won in less than 3 rounds >nited scores a goal in the first round*

?+)>nited won in less than 3 rounds and >nited scores a goal in the first round*

+)>nited scores a goal in the first round

?. .1 . .6 . .1

.

+

? .10

(ii) +)result is decided in at #ost nrounds* .6

1 +)result is not decided in nrounds* .6

+)result is not decided in nrounds* .0n

5

3 .0

n5

3ln

0.ln? 10.660

Deast n? 13

0 1

0 1

13 3 13 3 13 3 13.... .6

5 5 5 5 5 5 5

13 3 3 31 .... .6

5 5 5 5

13 31

5 5.6

31

5

3.0

5

n

n

n

n

+ + + + >

+ + + + > >

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Tut 30 POYO (Solutions)