Discrete Random Variables we are working with commonly used

Discrete Random Variables
we are working with commonly used discrete and continuous distribution

A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0.2 (independently of the other participants). What is the probability that for each group at least 9 participants complete the study?

Solution

3.
Here,

P(complete) = 1 - 0.2 = 0.8.

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 =    10      
p = the probability of a success =    0.8      
x = our critical value of successes =    9      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   8   ) =    0.624190362
          
Thus, the probability of at least   9   successes is  
          
P(at least   9   ) =    0.375809638   

Thus, if this happens to both groups,

P(both groups have at least 9 completers) = 0.375809638*0.375809638 = 0.141232884 [ANSWER]