RI(JC) Probability Tutorial Challenging Questions

RJC Probability 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.

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 18 17 16 15   12 11 10 9 8 7  10! P( R  4)                 0.0941  30 29 28 27   26 25 24 23 22 21  4!6!

(ii)

The most probable number of women on the committee is denoted by r. By using the fact that P(R = r) > P(R = r + 1), show that r satisfies the inequality (r + 1)!(17 – r)!(9 – r)!(r + 3)! > r!(18 – r)!(10 – r)!(r + 2)! and use this inequality to find the value of r.

P( R  r ) 

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Pr  12 P10 r 10!  30 P10 r !10  r !

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P( R  r  1) 

18

Pr 1  12 P9 r 10!  30 P10  r  1! 9  r !

P( R  r )  P( R  r  1) 18

Pr  12 P10 r 10!   30 P10 r !10  r !

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Pr 1  12 P9 r 10!  30 P10  r  1! 9  r !

18! 12! 1 18! 12! 1      18  r !  2  r ! r !10  r ! 17  r !  3  r !  r  1! 9  r ! 1 1 1 1 1 1      18  r !  2  r ! r !10  r ! 17  r !  3  r !  r  1! 9  r !

 r  1!17  r ! 9  r ! r  3!  r !18  r !10  r ! r  2 ! (shown)  r  1 r !17  r ! 9  r ! r  3 r  2 !  r !18  r 17  r !10  r 9  r ! r  2 !  r  1 r  3  18  r 10  r  r 2  4r  3  180  28r  r 2 32r  177 177  5.53125, i.e. r  6, 7, 8, 9, 10 32 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) r  6 r

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RJC Probability 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 k white balls, if balls are selected (i) with replacement, (ii) without replacement. (i)

If balls are selected with replacement,

k

 m   n  P(exactly k white balls)       mn  mn

r k

r! k ! r  k !

 r  m   n         k  m  n   m  n  k

(ii)

r k

If balls are selected without replacement,

P(exactly k white balls) k times

(r – k) times

 r  m  m  1 m  2  ...  m   k  1   n  n  1 n  2  ...  n   r  k  1     m  n  m  n  1 m  n  2  ...  m  n   r  1  k 

r times

 r  mP  nP    km  n r  k Pr k  m Pk  n Pr  k r!  k ! r  k ! m  n Pr m

n Pk Pr  k  k !  r  k !  m n Pr r!  m n     k  r  k    m  n    r 

n n nP Recall that   r !  n Pr , hence    r r  r  r!

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RJC Probability (b)

Balls are randomly selected one at a time until a white one is obtained, each ball being replaced before the next one is selected. Find the probability that (i) it will take exactly k draws, (i)

(ii) at least k draws are needed.

The probability that it will take exactly k draws,  n  P(it will take exactly k draws)     mn

k 1

 m     mn

draw black balls for the first (k – 1) times (ii)

The probability that at least k draws are needed, P(at least k draws needed) k draws (k + 1) draws (k + 2) draws  n     mn

k 1

k

 m   n   m   n         mn  mn  mn  mn k 1 k k 1   m   n   n   n           ...  m  n   m  n  mn mn 

k 1

 m     ... mn

k 1

 n     m   mn    m  n  1  n     mn  n    m  mn   m  mn mn  n     mn

sum to infinity of a GP

k 1

k 1

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RJC Probability An engineering system with n components is said to be a k out of n system if and only if at least k of 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 k out of n system. Similar to the previous ‘urn’ question, part (b),

P(exactly k components function)  p k 1  p 

nk

n! k ! n  k !

n nk    p k 1  p  k  P(at least k components function) k components (k + 1) components

(k + 2) components

n components

n  n  k 1  n  k 2 n nk n  k 1 n  k  2  nn    p k 1  p      ...    p n 1  p   p 1  p   p 1  p  k   k  1  k  2 n n n nr     p r 1  p  r k  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?

P(capturing 2 tagged deer out of 4) 

5 4 15 14 4! 70      20 19 18 17 2!2! 323

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