A grab bag contains 10 1 prizes 7 5 prizes and 5 20 prizes T

A grab bag contains 10 $1 prizes, 7 $5 prizes and 5 $20 prizes. Three prizes are chosen at random. Round to the nearest thousandth.

A. The probability that exactly two $20 prizes are chosen is

B. The probability that excatly one type of each are chosen is

C. The probability of picking at least one $20 prize is

D. The probablility that no $1 prizes are chosen is

Solution

a)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    22      
K = number of successes in the population =    5      
n = sample size =    3      
x = number of successes in the sample =    2      
          
Thus,          
          
P(   2   ) =    0.11038961 [ANSWER]

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b)

There are 10*7*5 = 350 ways to do it, and 22C3 ways to pick any 3.

Thus,

P = 350/(22C3) = 0.227272727 [ANSWER]

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c)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    22      
K = number of successes in the population =    5      
n = sample size =    3      
x = number of successes in the sample =    0      
          
Thus,          
          
P(   0   ) =    0.441558442

As P(at least 1) = 1 - P(0),

P(at least 1) = 0.558441558 [ANSWER]

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d)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    22      
K = number of successes in the population =    10      
n = sample size =    3      
x = number of successes in the sample =    0      
          
Thus,          
          
P(   0   ) =    0.142857143 [ANSWER]