Step by step
The question
Question: each try has a probability of success of 0.3: what is the probability of a first success within 3 tries?
p = 0.3: first success within 3 tries?
1 Name p and q
p, the chance of a success on one try, is 0.3.
p = 0.3
q, the chance of a fail, is 1 minus p: 0.7.
q = 1 − 0.3 = 0.7
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 3 tries
Within 3 means on try 1, or 2, up to 3.
The only other way: all 3 tries fail.
within 3: on try 1, or 2, ... or 3
the other way: all 3 fail
All 3 fail: q times itself 3 times.
That is 0.343.
all fail = q3 = (0.7)3 = 0.343
3 The rest
Everything else has a success within 3: 1 minus that is 0.6570.
P(X ≤ 3) = 1 − 0.343 = 0.6570
Careful!
p to the power 3 would be 3 successes in a row.
We only need one.
Careful: not 0.33 ✗
4 Check
Let's check by adding each try: 0.3000 plus 0.2100 plus 0.1470.
P(1) + ... + P(3) = 0.3000 + 0.2100 + 0.1470
That is about 0.6570.
So it is right!
≈ 0.6570
Let's look back.
First: name p and q.
Then: none in 3 tries.
Then: the rest.
Last: check.
Answer
Careful: not 0.33 ✗