Step by step
The question
Question: each try has a probability of success of 0.4: what is the probability of a first success within 2 tries?
p = 0.4: first success within 2 tries?
1 Name p and q
p, the chance of a success on one try, is 0.4.
p = 0.4
q, the chance of a fail, is 1 minus p: 0.6.
q = 1 − 0.4 = 0.6
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 2 tries
Within 2 means on try 1, or 2, up to 2.
The only other way: all 2 tries fail.
within 2: on try 1, or 2, ... or 2
the other way: all 2 fail
All 2 fail: q times itself 2 times.
That is 0.36.
all fail = q2 = (0.6)2 = 0.36
3 The rest
Everything else has a success within 2: 1 minus that is 0.6400.
P(X ≤ 2) = 1 − 0.36 = 0.6400
Careful!
p to the power 2 would be 2 successes in a row.
We only need one.
Careful: not 0.42 ✗
4 Check
Let's check by adding each try: 0.4000 plus 0.2400.
P(1) + ... + P(2) = 0.4000 + 0.2400
That is about 0.6400.
So it is right!
≈ 0.6400
Let's look back.
First: name p and q.
Then: none in 2 tries.
Then: the rest.
Last: check.
Answer
Careful: not 0.42 ✗