# Probability - Partition TheoremWatch

Announcements
This discussion is closed.
#1
A bag contains three balls - one red, one white, one orange.

A ball is chosen at random and replaced.

If the ball is white, the player receives two counters.

If the ball is orange, the player receives one counter.

If the ball is red, the player receives no counters.

What is the expected number of counters that the player will gain before choosing the red ball for the first time?
0
14 years ago
#2
Let X be the number of balls chosen up to and including the first red one. Since X ~ Geometric(1/3), E(X) = 3.

Let C be the number of counters gained from the first X draws.

Then

E(C | X = x) = (3/2)(x - 1)

because, conditional on X = x, each of the first x - 1 balls is independently white (with probability 1/2) or orange (with probability 1/2).

E(C)
= (sum over x) E(C | X = x)P(X = x)
= (sum over x) (3/2)(x - 1)P(X = x)
= (3/2)(E(X) - 1)
= 3
0
X
new posts
Back
to top
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### University open days

• University of Bristol
Wed, 23 Oct '19
• University of Exeter
Wed, 23 Oct '19
• University of Nottingham
Wed, 23 Oct '19

### Poll

Join the discussion

Yes I know where I'm applying (144)
60.25%
No I haven't decided yet (54)
22.59%
Yes but I might change my mind (41)
17.15%