Bipin's choice
Problem
Bipin is playing in a game show. The host tells Bipin that there are 10 balls hidden in a box - 5 are Red, 3 are Yellow and 2 are Blue. Bipin pulls out his first ball. It's Red and he wins a large sum of money. Now his choices are:
- He can stop playing and take his prize.
- He can pull out another ball - if it's Red he doubles his money, if it's Yellow he leaves with nothing, if it's Blue he can pick one last ball.
What should Bipin do? Why?
Getting Started
What is the chance of picking another red ball?
What is the chance of of picking a yellow ball?
Student Solutions
This is the kind of problem where it helps to have someone to talk about your ideas with. Catherine, Ruyi, Elle and Ebony from Annesley College, Adelaide, Australia worked in pairs and decided that although Bipin was taking a risk, he should choose to play again. Here's one of their reasons:
Bipin should play again because he has a four in nine chance of getting a red ball (this doubles his money). He has a three in nine chance of getting a yellow ball (this makes him lose all of his money) and a two in nine chance of getting a blue ball (this gives him another chance of drawing another ball).
This means that he has a greater chance of doubling his money. (Catherine)
Children at Carlingford West Primary School in Sydney, Australia, had a go at this task. Marcus says:
If the game is played with Bipin allowed to put the ball back in, then I think that he should choose another ball because there is a $\frac{7}{10}$ chance that it will be a good ball as the only bad ball is the yellow ball. It is a $\frac{7}{10}$ good event because $\frac{5}{10}$ red being selected and $\frac{2}{10}$ of selecting blue. That is why he should pick another ball from the box.
If the ball is not allowed to be put back in the box, then the number of red and blue will decrease the chance of picking the good balls and each turn if Bipin picks a good ball, the chance changes from $\frac{7}{10}$ to $\frac{6}{9}$. I still think that he should pick another ball because the chance is higher to get a good result and not a bad result.
However, Ruby says:
Because he has already got one red ball and a large sum of money, the chance that he could get another red ball gets lower because he already has one. If he gets yellow, he gets nothing and he can’t even take the prize money with the first red ball. He can take one last ball and if it was blue, I think that would leave him with nothing because the text said ‘one last ball’ and if he gets yellow, he gets nothing. The chance of getting a red ball is very low so he should stop playing and take his prize.
Ruby added this image to help:
This is her explanation of the diagram:
In the top left hand corner you can see a blue box with the word ‘before’ next to it. In that box there are 5 red balls, 3 yellow balls and 2 blue balls. That makes 10 balls. This is how it started before Bipin got 1 red ball out.
Beneath that box is another box with the word ‘after’ next to it. This is after Bipin chose 1 red ball. Now there are 4 red balls, 3 yellow balls and 2 blue balls.
Next to all that is another diagram which has glowing letters after at the top. After Bipin got 1 red ball out, you can see that blue has a chance of 2 out of 9, yellow has a chance of 3 out of 9 and red has a chance of now 4 out of 9. Although red has the highest chance to get it, there is still a possibility that he could get a yellow ball, so that’s why I chose STOP.
Thank you to both Marcus and Ruby - what fantastic reasoning from them both. Drawing a diagram or picture can often be a very useful way of helping you solve a problem.
What do you think? Do you agree with Catherine, Ruyi, Elle, Ebony and Marcus and think that Bipin should play again, or do you agree with Ruby and think he should take his prize? Let us know - and please tell us how you reached your conclusion.