a)

First attempt

-------Pass
------/
-----/
----/
---/ 3/5
--/
-/
Test
-\
--\
---\
----\ 2/5
-----\□□□□□□□second attempt
------\
-------\
--------\-----------------Pass
---------\--------------/
-----------\----------/ 3/5
------------\-------/
------------- Fail
-----------------\
-------------------\ 2/5
---------------------\
-----------------------Fail

b)

The phrasing is odd so here are two solutions :


Interpretation 1 : taking it to mean passing on the 2nd attempt

Probability of failing = 1 - 3/5 = 2/5

Probability of getting a certificate on first 2 attempts

= Probability of failing first attempt x probability of passing second attempt


= 2/5 x 3/5
= 6/25



Interpretation 2 : taking it to mean passing on either the 1st or 2nd attempt.

Probability of failing = 1 - 3/5 = 2/5

Probability of getting a certificate on 2nd attempt

= Probability of failing first attempt x probability of passing second attempt

= 2/5 x 3/5
= 6/25


Probability of getting a certificate on first attempt = 3/5


Total probability
= 3/5 + 6/25
= 15/25 + 6/25
= 21/25

c)

Interpretation 1 :

The 3 participants took 2 attempts each but only 1 made it by the second attempt.

That means the other 2 failed both attempts.

For the two that failed, probability of failing both = 2/5 x 2/5 = 4/25

For the one that passed, the probability
= 6/25 (same as part b 1st interpretation)

Required Probability

= 3 x 4/25 x 4/25 x 6/25

= 288/15625


(Multiplying by 3 is needed as you could have the following situations :

Student 1 passes , other 2 fails
Student 2 passes , other 2 fails
Student 3 passes , other 2 fails




Interpretation 2 :

Includes the above probability and a second case where only one gets a cert but on the first attempt.

For this second case,


For the two that failed, probability of failing both = 2/5 x 2/5 = 4/25

For the one that passed, the probability
= 3/5


Probability for this case
= 3 x 4/25 x 4/25 x 3/5

= 144/3125


Required probability

= 288/15625 + 144/3125

= 288/15625 + 720/15625

= 1008/15625

Omg, you succeeded at drawing a tree diagram on text!

Inspired by snell

d)

Passing one of the first n attempts means that n attempts actually have to be made. If not the person would have already attained the cert earlier.

This means the first (n - 1) attempts were fails and the last is a pass.



If we fail once, it's 2/5 x 3/5

If we fail twice, it's 2/5 x 2/5 x 3/5
= (2/5)² x 3/5

If we fail thrice it's 2/5 x 2/5 x 2/5 x 3/5
= (2/5)³ x 3/5


So for failing (n - 1) times , we end up having 2/5 to the power of (n - 1).


Required probability


= (2/5)ⁿ‾¹ x (3/5)

= (3/5)(2/5)ⁿ‾¹

I took it that “on the first two attempts” meant either of the first two.

Maybe you are right, it could have meant “on the second attempt”.

The phrasing of the question appears misleading.

Missed that the nth attempt, and not the n + 1 th attempt was a pass.

Chelsea, follow J’s solutions for this question. I have removed my solutions accordingly.

Think I better write both solutions