Step by step
The question
Question: each try has a probability of success of 0.5: what is the probability of a first success within 4 tries?
p = 0.5: first success within 4 tries?
1 Name p and q
p, the chance of a success on one try, is 0.5.
p = 0.5
q, the chance of a fail, is 1 minus p: 0.5.
q = 1 − 0.5 = 0.5
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 4 tries
Within 4 means on try 1, or 2, up to 4.
The only other way: all 4 tries fail.
within 4: on try 1, or 2, ... or 4
the other way: all 4 fail
All 4 fail: q times itself 4 times.
That is 0.0625.
all fail = q4 = (0.5)4 = 0.0625
3 The rest
Everything else has a success within 4: 1 minus that is 0.9375.
P(X ≤ 4) = 1 − 0.0625 = 0.9375
Careful!
p to the power 4 would be 4 successes in a row.
We only need one.
Careful: not 0.54 ✗
4 Check
Let's check by adding each try: 0.5000 plus 0.2500 plus 0.1250 plus 0.0625.
P(1) + ... + P(4) = 0.5000 + 0.2500 + 0.1250 + 0.0625
That is about 0.9375.
So it is right!
≈ 0.9375
Let's look back.
First: name p and q.
Then: none in 4 tries.
Then: the rest.
Last: check.
Answer
Careful: not 0.54 ✗