Picking the same ball 3 times

[maheen]

Hi, I am having so much difficulty with this problem I don't even know where to start.

You have 3 balls in a bag, 2 black and 1 red. 1 ball is picked at random, the colour is noted and the ball is put back in the bag.
2 black balls are added in the bag making a total of 5 balls. 1 ball is picked at random, the colour is noted and the ball is put back in the bag.
2 more black balls are added in the bag making a total of 7 balls. 1 ball is picked at random, the colour is noted and the ball is put back in the bag.

what is the probability of the red ball being picked on all three occasions?

Please help
Much much appreciated, thank you.

 

[Dr.Peterson]

Hi. We can help you more effectively if you show us whatever work or ideas you have.

I would start by just working out the first part:

You have 3 balls in a bag, 2 black and 1 red. 1 ball is picked at random, the colour is noted and the ball is put back in the bag.​

What is the probability that this ball is red?

Then we can continue from there.

 

[pka]

Note on notation: \(X_{n}=\text{colour of the ball on the }n^{th}\text{ draw}\)
You want \(\mathcal{P}\left(X_1=\text{red}\right)\cdot\mathcal{P}\left(X_2=\text{red}\right)\cdot\mathcal{P}\left(X_3=\text{red}\right)\).
Do not forget the ball in returned to the bag along with two additional black balls.

 

[maheen]

so does this make any sense:

3 successes
on each draw the probability changes, n=1: 1/3, n=2: 1/5, n=3: 1/7

so probability = 3⋅(1/3)⋅(1/5)⋅(1/7) = 1/35

Is this correct

 

[pka]

so does this make any sense:
3 successes
on each draw the probability changes, n=1: 1/3, n=2: 1/5, n=3: 1/7
so probability = 3⋅(1/3)⋅(1/5)⋅(1/7) = 1/35
Is this correct

Why have a factor of 3?
These are three independent events the probability that all thee occur is simply the product of each.

 

[maheen]

Why have a factor of 3?
These are three independent events the probability that all thee occur is simply the product of each.

Ah yes, I see it now -
the answer would be:
(1/3)⋅(1/5)⋅(1/7) = 1/105

thank you all for your help.