Probability of success 0.3, P(first success on try 4) Step by Step

Step by step

The question

Question: each try has a probability of success of 0.3: what is the probability that the first success comes on try 4?

p = 0.3: first success on try 4?

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 0.3.

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 Fail, then win

First success on try 4: the first 3 tries fail, then try 4 wins.

tries 1 to 4: F F F S

F = fail, S = success

Fail, fail, and so on, then win: we multiply.

q for each of the 3 fails, then p for the win.

P(X = 4) = q × q × q × p = 0.7 × 0.7 × 0.7 × 0.3

Careful!

Only 3 fails come before the win, not 4.

Careful: not q⁴ × p ✗

3 Multiply

0.7 times 0.7 times 0.7 is 0.343.

Times 0.3: 0.1029.

P(X = 4) = 0.343 × 0.3 = 0.1029

4 Check

Let's check with 10000 people trying.

On each try, 30 percent of those left win, the rest go on.

Try 1: 3000 win, 7000 go on.

Try 2: 2100 win, 4900 go on.

Try 3: 1470 win, 3430 go on.

10000 people: try 1: 3000 win, 7000 go on; try 2: 2100 win, 4900 go on; try 3: 1470 win, 3430 go on;

On try 4, 1029 win for the first time.

1029 out of 10000 is about 0.1029.

So it is right!

try 4: 1029 win: 1029 ÷ 10000 ≈ 0.1029

Let's look back.

First: name p and q.

Then: fail, then win.

Then: multiply.

Last: check.

Answer