Probability of success 0.5, P(first success on try 2) Step by Step

Step by step

The question

Question: each try has a probability of success of 0.5: what is the probability that the first success comes on try 2?

p = 0.5: first success on try 2?

X counts the tries up to the first success.

X = the try of the first success

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 Fail, then win

First success on try 2: the first 1 try fail, then try 2 wins.

tries 1 to 2: F S

F = fail, S = success

Fail, then win: we multiply.

q for each of the 1 fails, then p for the win.

P(X = 2) = q × p = 0.5 × 0.5

Careful!

Only 1 fails come before the win, not 2.

Careful: not q² × p ✗

3 Multiply

0.5 is 0.5.

Times 0.5: 0.2500.

P(X = 2) = 0.5 × 0.5 = 0.2500

4 Check

Let's check with 100 people trying.

On each try, 50 percent of those left win, the rest go on.

Try 1: 50 win, 50 go on.

100 people: try 1: 50 win, 50 go on;

On try 2, 25 win for the first time.

25 out of 100 is about 0.2500.

So it is right!

try 2: 25 win: 25 ÷ 100 ≈ 0.2500

Let's look back.

First: name p and q.

Then: fail, then win.

Then: multiply.

Last: check.

Answer