ri(jc) probability tutorial challenging questions

8/2/2019 RI(JC) Probability Tutorial Challenging Questions

1/4

RJC Probability

1

A committee of 10 people is chosen at random from a group consisting of 18 women and 12 men.

The number of women on the committee is denoted by R.

(i) Find the probability that R = 4. [3]

18 17 16 15 12 11 10 9 8 7 10!P( 4) 0.0941

30 29 28 27 26 25 24 23 22 21 4!6!R

(ii) The most probable number of women on the committee is denoted by r. By using the factthat P(R = r) > P(R = r+ 1), show that rsatisfies the inequality

(r+ 1)!(17r)!(9r)!(r+ 3)! > r!(18r)!(10r)!(r+ 2)!

and use this inequality to find the value ofr. [5]

18 12 18 12

10 1 9

30 30

10 10

P P P P10! 10!P( ) P( 1)

P ! 10 ! P 1 ! 9 !

r r r r R r R r r r r r

18 12 18 12

10 1 9

30 30

10 10

P( ) P( 1)

P P P P10! 10!

P ! 10 ! P 1 ! 9 !

18! 12! 1 18! 12! 1

18 ! 2 ! ! 10 ! 17 ! 3 ! 1 ! 9 !

1 1 1 1 1 1

18 ! 2 ! ! 10 ! 17 ! 3 ! 1 ! 9 !

1 ! 17 ! 9 ! 3 ! ! 18 ! 10 !

r r r r

R r R r

r r r r

r r r r r r r r

r r r r r r r r

r r r r r r r

2 ! (shown)r

2 2

1 ! 17 ! 9 ! 3 2 ! ! 18 17 ! 10 9 ! 2 !

1 3 18 10

4 3 180 28

32 177

1775.53125, i.e. 6, 7, 8, 9, 10

32

r r r r r r r r r r r r

r r r r

r r r r

r

r r

Since P(R = r) > P(R = r+ 1) (based on the question),for r= 6, P(R = 6) > P(R = 7) > P(R = 8) > P(R = 9) > P(R = 10)

6r


2/4

RJC Probability

2

An urn contains m white balls and n black balls.

(a) If a random sample of size r is chosen, what is the probability that it contains exactly kwhite balls, if balls are selected (i) with replacement, (ii) without replacement.

(i) If balls are selected with replacement,

!

P(exactly white balls)! !

k r k

k r k

m n rk

m n m n k r k

r m n

k m n m n

(ii) If balls are selected without replacement,

P(exactly white balls)

1 2 ... 1 1 2 ... 1

1 2 ... 1

P P

P

P P!

! ! P

P P

! !Recall that ! P ,

P

!

m n

k r km n

r

m n

k r k

m n

r

m n

k r k

n

rm n

r

k

m m m m k n n n n r k r

k m n m n m n m n r

r

k

r

k r k

nk r kr

r

r

Phence

!

n

rn

r r

m n

k r k

m n

r

k times (rk) times

r times


3/4

RJC Probability

3

(b) Balls are randomly selected one at a time until a white one is obtained, each ball beingreplaced before the next one is selected. Find the probability that

(i) it will take exactly kdraws, (ii) at least kdraws are needed.

(i) The probability that it will take exactly kdraws,1

P(it will take exactly draws)

kn m

km n m n

(ii) The probability that at least kdraws are needed,

1 1

1 1

1

P(at least draws needed)

...

...

1

k k k

k k k

k

k

n m n m n m

m n m n m n m n m n m n

m n n n

m n m n m n m n

n

m m n

nm n

m n

m

1

1

k

k

nm n

mm n

m n

n

m n

draw black balls for the first (k1) times

k draws (k + 1) draws (k + 2) draws

sum to infinity of a GP


4/4

RJC Probability

4

An engineering system with n components is said to be a kout ofn system if and only if at least

kof the components function. Suppose that all the components function independently of each

other. Given that the probability that each component functions is p, compute the probability that

the system is a kout ofn system.

Similar to the previous urn question, part (b),

!P(exactly components function) 1

! !

1

n kk

n kk

nk p p

k n k

np p

k

1 21 2

P(at least components function)

1 1 1 ... 11 2

1

n k n k n k n nk k k n

nn rr

r k

k

n n n n

p p p p p p p pk k k n

np p

r

A forest contains 20 deer, of which 5 are captured, tagged, and then released. A certain time later,

4 of the 20 deer are captured. What is the probability that 2 of these 4 have been tagged?

5 4 15 14 4! 70P(capturing 2 tagged deer out of 4)

20 19 18 17 2!2! 323

k components (k + 1) components (k + 2) components n components

ri(jc) probability tutorial challenging questions

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