Please show work A jar contains 4 yellow 3 orange and 7 red

Please show work.

A jar contains 4 yellow, 3 orange, and 7 red jelly beans. Three jelly beans are selected at random. Find the probability that the selection includes the following. All orange jelly beans 3 yellow and 2 red jelly beans At most 1 yellow jelly bean At least 2 red jelly beans

Solution

There are a total of 14 beans here.

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 =    14      
K = number of successes in the population =    3      
n = sample size =    3      
x = number of successes in the sample =    3      
          
Thus,          
          
P(   3   ) =    0.002747253 [ANSWER]

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b)

0. [ANSWER]

This is impossible. You only get 3 beans.

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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 =    14      
K = number of successes in the population =    4      
n = sample size =    3      
x = number of successes in the sample =    0      
          
Thus,          
          
P(   0   ) =    0.32967033

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

Then

P(at least 1 yellow) = 1 - 0.32967033 = 0.67032967 [ANSWER]

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d)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative hypergeometric distribution table or technology, matching          
          
where          
N = population size =    14      
K = number of successes in the population =    7      
n = sample size =    3      
x = critical number of successes in the sample =    2      
          
Thus,          
P(at most   1   ) =    0.5
Thus, the probability of at least   2   successes is  
          
P(at least   2   ) =    0.5 [ANSWER]