Probability of success 0.1, P(first success on try 3) Step by Step

Step by step

The question

Question: each try has a probability of success of 0.1: what is the probability that the first success comes on try 3?

p = 0.1: first success on try 3?

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 0.1.

p = 0.1

q, the chance of a fail, is 1 minus p: 0.9.

q = 1 − 0.1 = 0.9

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 3: the first 2 tries fail, then try 3 wins.

tries 1 to 3: F F S

F = fail, S = success

Fail, fail, then win: we multiply.

q for each of the 2 fails, then p for the win.

P(X = 3) = q × q × p = 0.9 × 0.9 × 0.1

Careful!

Only 2 fails come before the win, not 3.

Careful: not q³ × p ✗

3 Multiply

0.9 times 0.9 is 0.81.

Times 0.1: 0.0810.

P(X = 3) = 0.81 × 0.1 = 0.0810

4 Check

Let's check with 1000 people trying.

On each try, 10 percent of those left win, the rest go on.

Try 1: 100 win, 900 go on.

Try 2: 90 win, 810 go on.

1000 people: try 1: 100 win, 900 go on; try 2: 90 win, 810 go on;

On try 3, 81 win for the first time.

81 out of 1000 is about 0.0810.

So it is right!

try 3: 81 win: 81 ÷ 1000 ≈ 0.0810

Let's look back.

First: name p and q.

Then: fail, then win.

Then: multiply.

Last: check.

Answer