Last week I went to a state fair which I saw a game of lucky draw. There is two sealed boxes, contains bunch of 4 different color balls: red, blue, green and white. Here is the game rule. Players make an initial draw on box one, if players get a white ball, lose the game; if getting a red one, get 4 free draws; getting a blue one, get 3 free draws; getting a green one, get 2 free draws. Players make the free draws in the second box. In this scenario, in any draw if players getting any red ball, 3 more draws added; any blue one, 2 more draws added; getting any green one, 1 more draw added; getting white one does not award additional drawing chance but will NOT lose the game immediately. The game is over when all drawing chances used up. The reward gift based on how many total color balls are obtained. I am interesting in finding out how many draws in average will this game given in case a non-white ball was drawn in first round?(adsbygoogle = window.adsbygoogle || []).push({});

I did some math based on what I learnt from the text as follows. Let's assume for the first draw, the chance to get a red/blue/green ball is ##p_1=0.05, p_2=0.1## and ##p_3=0.17## respectively so 1-0.05-0.1-0.17=0.68 chance will get a white one and lose the game.

After the first draw, if not losing the game, the chance to get a red/blue/green ball is ##q_1=0.01, q_2=0.03## and ##q_3=0.045## respectively. So in this scenario, the contribution to the average time from the initial win 4/3/2 is

##\displaystyle E_4 = \frac{4}{1-(q_1\times3 + q_2\times 2 + q_1\times 1)};##

##\displaystyle E_3 = \frac{3}{1-(q_1\times3 + q_2\times2 + q_1\times 1)};##

##\displaystyle E_2 = \frac{2}{1-(q_1\times 3 + q_2\times 2 + q_1\times 1)}.##

But note that there is ##p_1/p_2/p_3## chance to get 4/3/2 draws initially. So the total average number of draws should be

##

\overline{D} = p_1 E_4 + p_2 E_3 + p_3 E_2

##

By plugging the numbers into that, I got 3.035 draws. I am not quite sure this is correct because I write a short program to simulate this game but I get the average draw to be 1.05 instead.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Statistical problem of drawing colored balls from boxes

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Statistical problem drawing | Date |
---|---|

I Advanced problems with answer sheet in statistics | Jul 27, 2017 |

B Sweets in a bag probability problem | May 28, 2017 |

A Stopping rule for quality control problem | Sep 2, 2016 |

Shuffling Songs probability question | Apr 19, 2015 |

**Physics Forums - The Fusion of Science and Community**