When applicable please round answers to the thousandth Angel

When applicable, please round answers to the thousandth** Angela has a 6-sided number cube with sides labeled 1 through 6. If she rolls the cube twice, what is the probability that the product of the two rolls is less than 30? Suppose that the probability of event \"green eyes\" is currently p = .35 for children born in the United States and q = .65 for event \"non-green eyes.\" If a sample of 15 births is taken at random (independent births, so that twins, and so on. are excluded.), then.... What is the probability of fewer than 5 children with green eyes? What is the probability of fewer than four or of more than 8, green-eyed children out of 15 births?

Solution

1.

Note that there are 6*6 = 36 possible outcomes.

There are only 3 possibilties that have a product that is not less than 30: (5,6), (6,5), and (6,6).

Thus, there are 36 - 3 = 33 products less than 30.

Thus,

P = 33/36 = 11/12 or 0.916666667 [ANSWER]

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2.

a)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.35      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.351943428
          
Which is also          
          
P(fewer than   5   ) =    0.351943428 [ANSWER]

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b)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.35      
x = our critical value of successes =    8      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   8   ) =    0.957806162
          
Thus, the probability of at least   9   successes is  
          
P(more than   8   ) =    0.042193838

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    15      
p = the probability of a success =    0.35      
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.172696487
          
Which is also          
          
P(fewer than   4   ) =    0.172696487

Thus,

P(more than 4 or more than 8) = 0.042193838 + 0.172696487 = 0.214890325 [ANSWER]