Math Help for High School. Calculations.

Question

The problem is: Ann is in charge of a Lucky Dip to raise money for charities. Each barrel contains an equal number of red, green, white and black balls. The balls are buried in sawdust so that you cannot see them before you pick one out. To play the game, you give Ann your 25 cents, then you pick one ball from each barrel. You win 5 dollars if all three balls are the same color. Questions: 1. Calculate the probability that you will win the $5 if you play once.

2. Do you think that the Lucky Dip will raise money for the local charities?

I did not made any attempts to do it for I don't know how to solve this problem. I think that the solution may include either division or multiplication.

Answer 1

Major hints and setup

$Pr(\text{All are same}) = Pr(\text{all are red})+Pr(\text{all are green})+Pr(\text{all are white})+Pr(\text{all are black})$

Now, let us look at one of these in more detail. It is heavily implied that the selections are independent, so we have

$Pr(\text{all three are red}) = Pr(\text{first is red})\cdot Pr(\text{second is red})\cdot Pr(\text{third is red})$

Now... what is $Pr(\text{first is red})$? There are an equal number of red, green, white, and black balls in the first barrel. What then is the probability of picking a red ball?

$Pr(\text{first is red})=\frac{1}{4}$

What is the probability of the second ball being red? Of the third ball? What then is the probability that all three are red?

$Pr(\text{all three are red})=\frac{1}{4}\cdot \frac{1}{4}\cdot \frac{1}{4}$

How does this change when we look instead to finding $Pr(\text{all three are green})$ or $Pr(\text{all three are white})$? What then is the probability $Pr(\text{all three are the same color})$?

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Now, we ask if they will make money or lose money. The charity earns $\$0.25$ if the player loses and will have a net loss of $\$4.75$ if the player wins. What is the probability that the player wins? What is the probability that the player loses?

Where $X$ is the amount of money earned by the charity in a game, we have the expected value of winnings for the charity to be $E[X] = 0.25\cdot Pr(X=0.25)+(-4.75)\cdot Pr(X=-4.75) = 0.25\cdot Pr(\text{player loses})+(-4.75)Pr(\text{player wins})$

What is $Pr(\text{player wins})$? How does that relate to $Pr(\text{all three are the same color})$? What then is $Pr(\text{player loses})$?

What then is $E[X]$? Is it positive or negative? Do we expect that the charity then will make money or lose money?