Step by step
The question
Question: each try has a probability of success of 0.2: what is the probability of a first success within 5 tries?
p = 0.2: first success within 5 tries?
1 Name p and q
p, the chance of a success on one try, is 0.2.
p = 0.2
q, the chance of a fail, is 1 minus p: 0.8.
q = 1 − 0.2 = 0.8
Every try has the same chance, and one try does not change the next.
every try: the same p, and the tries do not affect each other
2 None in 5 tries
Within 5 means on try 1, or 2, up to 5.
The only other way: all 5 tries fail.
within 5: on try 1, or 2, ... or 5
the other way: all 5 fail
All 5 fail: q times itself 5 times.
That is 0.3277.
all fail = q5 = (0.8)5 = 0.3277
3 The rest
Everything else has a success within 5: 1 minus that is 0.6723.
P(X ≤ 5) = 1 − 0.3277 = 0.6723
Careful!
p to the power 5 would be 5 successes in a row.
We only need one.
Careful: not 0.25 ✗
4 Check
Let's check by adding each try: 0.2000 plus 0.1600 plus 0.1280 plus 0.1024 plus 0.0819.
P(1) + ... + P(5) = 0.2000 + 0.1600 + 0.1280 + 0.1024 + 0.0819
That is about 0.6723.
So it is right!
≈ 0.6723
Let's look back.
First: name p and q.
Then: none in 5 tries.
Then: the rest.
Last: check.
Answer
Careful: not 0.25 ✗