Question

A bag contains 5 green balls and 3 yellow balls. Ronnie and Julie play a game in which they take turns to draw a ball from the bag at random without replacement. The winner of the game is the first person to draw a yellow ball. Julie draws the first ball. Find the probability that Ronnie wins the game.

Answer 1

this is a tricky one

we have got \( \color{green}{5} \) green balls, and \( \color{yellow}{3} \) yellow balls

lets say that we are ronnie

whoever pulls a yellow ball out first, is the winner.

since julie draws first, logically we are gonna be pulling after her, and then she will be drawing another ball again, and so will us, and so on.

what is the chance of US winning the game? there are a couple of options which may happen

(g \(\rightarrow \) green ; y \(\rightarrow \) yellow )

1) she draws a green and we will draw a yellow right after \( \Rightarrow P(gy) \)

2) she draws a green and we will draw a a green as well, and she again draws a green and we pull out a yellow \( \Rightarrow P(gggy) \)

3) she draws a green and we will draw a green as well, and she again draws a green and we draw a green again, and she draws a green after that, and here are no more green balls left, so all its left is to pull out one of the left 3 remaining yellow balls \( \Rightarrow P(gggggy) \)

Answer 2

there are 8 balls in total, so whener a ball is pulled out, we take 1 off the denominator

whenever we will be pulling a yellow ball its always \( \frac{3}{number} \) since we've done it and we won already -- so off we start with the fractions:

1) \( P(gy) = \frac{5}{8} \) (thats the resoluteness of that first pull) \( \times \frac{3}{7} \)

2) \( P(gggy) = \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{3}{5} \)

3) \( P(gggggy) = \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4} \times \frac{3}{3} \)

Answer 3

the chance of us winning is the sum of all of these -- so that means

\[ P( \text {us winning} ) = P(gy) + P(gggy)+P(gggggy) \]
\[ \ \ \ \\ \ \ \ \\ \ \ \\ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \Updownarrow \]
\[ P( \text {us winning} ) = \Bigg( \frac{5}{8} \times \frac{3}{7} \Bigg ) + \Bigg( \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{3}{5} \Bigg) + \Bigg( \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4} \times \frac{3}{3} \Bigg)\]

all its left to do, it solve it, and you will have an answer.

Answer 4

\( \Bigg( \frac{5}{8} \times \frac{4}{7} \times \frac{3}{6} \times \frac{2}{5} \times \frac{1}{4} \times \frac{3}{3} \Bigg) \)

oopsies, it was too long -- thats the last one