12th Grade 3-part probability problem.

Hi,

Im currently studying 12th grade advanced mathematics and have come across this rather difficult probability question. It has three parts. I'm struggling in particular with parts ii) and iii). Could somebody please provide worked solutions as i unfortunately do not have answers to this problem. Any help will be much appreciated.

Question:

There are 4 red discs and 3 black discs in bag. Sophie and Emma play a game where they each have a turn at drawing a disc with replacement. Sophie starts.

Sophie wins when she draws a red disc and Emma wins when she draws a black disc.

(i) What's the probability that Sophie wins on the first draw?

(ii) What is the probability that Sophie wins in the first 3 draws. (either 3 or less)

(iii) What is the probability that Sophie wins the game

Appreciate any help :)

Re: 12th Grade 3-part probability problem.

Have you tried drawing a tree diagram?

Re: 12th Grade 3-part probability problem.

Quote:

Originally Posted by

**Pinchy444** Question:

There are 4 red discs and 3 black discs in bag. Sophie and Emma play a game where they each have a turn at drawing a disc with replacement. Sophie starts.

Sophie wins when she draws a red disc and Emma wins when she draws a black disc.

(i) What's the probability that Sophie wins on the first draw?

(ii) What is the probability that Sophie wins in the first 3 draws. (either 3 or less)

(iii) What is the probability that Sophie wins the game.

Does "Sophie wins in the first 3 draws" either on her first or her second draw? You see she draws, Emma draws, and Sophie draws again. That is three draws. Or does it means she wins her first, her second, or her third draw?

If we use the second meaning (within her third) the answer is: $\displaystyle \left(\frac{4}{7}\right)+\left(\frac{3}{7} \right)\left(\frac{4}7}\right)^2+\left(\frac{3}{7} \right)^2\left(\frac{4}{7}\right)^3$.

Re: 12th Grade 3-part probability problem.

Quote:

Originally Posted by

**Prove It** Have you tried drawing a tree diagram?

I don't think a tree diagram is absolutely necessary, though it COULD help.

(i) The probability she wins on first draw is 4/7

(ii) Eh, I messed up

(iii) List a few out and you should notice a recursion pattern.