Answer:
a) Our random variable X="number of tests taken until the individual passes" follows a geomteric distribution with probability of success p=0.25
For this case the probability mass function would be given by:

b) 



And adding the values we got:

c) 
And we can find the individual probabilities:




![P(X \geq 5) = 1-[0.25+0.1875+0.1406+0.1055]= 0.316](https://tex.z-dn.net/?f=%20P%28X%20%5Cgeq%205%29%20%3D%201-%5B0.25%2B0.1875%2B0.1406%2B0.1055%5D%3D%200.316)
Step-by-step explanation:
Previous concepts
The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"
Let X the random variable that measures the number of trials until the first success, we know that X follows this distribution:
Part a
Our random variable X="number of tests taken until the individual passes" follows a geomteric distribution with probability of success p=0.25
For this case the probability mass function would be given by:

Part b
We want this probability:

We find the individual probabilities like this:



And adding the values we got:

Part c
For this case we want this probability:

And we can use the complement rule like this:

And we can find the individual probabilities:




![P(X \geq 5) = 1-[0.25+0.1875+0.1406+0.1055]= 0.316](https://tex.z-dn.net/?f=%20P%28X%20%5Cgeq%205%29%20%3D%201-%5B0.25%2B0.1875%2B0.1406%2B0.1055%5D%3D%200.316)