Probability of success 0.6, P(first success on try 3) Step by Step

Step by step

The question

Question: each try has a probability of success of 0.6: what is the probability that the first success comes on try 3?

p = 0.6: 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 0.6.

p = 0.6

q, the chance of a fail, is 1 minus p: 0.4.

q = 1 − 0.6 = 0.4

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 = 0.4 × 0.4 × 0.6

Careful!

Only 2 fails come before the win, not 3.

Careful: not q³ × p ✗

3 Multiply

0.4 times 0.4 is 0.16.

Times 0.6: 0.0960.

P(X = 3) = 0.16 × 0.6 = 0.0960

4 Check

Let's check with 1000 people trying.

On each try, 60 percent of those left win, the rest go on.

Try 1: 600 win, 400 go on.

Try 2: 240 win, 160 go on.

1000 people: try 1: 600 win, 400 go on; try 2: 240 win, 160 go on;

On try 3, 96 win for the first time.

96 out of 1000 is about 0.0960.

So it is right!

try 3: 96 win: 96 ÷ 1000 ≈ 0.0960

Let's look back.

First: name p and q.

Then: fail, then win.

Then: multiply.

Last: check.

Answer