suppose 20 of a population consists of people with blonde ha

suppose 20% of a population consists of people with blonde hair. 8 people are randomly selected from this population.

a) find the probability that exactly 2 of them have blonde hair

b)Find the probability that at least 1 of them have blonde hair

c)find the probability that at least 6 of them have blonde hair.

Solution

A)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    8      
p = the probability of a success =    0.2      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.29360128 [ANSWER]

********************

b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.2      
x = our critical value of successes =    1      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   0   ) =    0.16777216
          
Thus, the probability of at least   1   successes is  
          
P(at least   1   ) =    0.83222784 [ANSWER]

***********************

c)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    8      
p = the probability of a success =    0.2      
x = our critical value of successes =    6      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   5   ) =    0.99876864
          
Thus, the probability of at least   6   successes is  
          
P(at least   6   ) =    0.00123136 [ANSWER]