Question: 20% of gumballs in a machine are red. The other 80% are blue. 20% of the gumballs in the machine have dents in them. What is the probability of getting a dented gumball that is red?
My Approach: We can call the probability of a red gumball P(r), the probability of a blue gumball P(b) and the probability of a dented gumball P(d).
We know P(r) and P(b) are independent as you can't have gumballs that are both red and blue.
The equation I want to use is P(r|d): (Join probability of r & d)/P(d)
So given that the gumball is dented, what is the likelihood it is red.
P(d) = .2 P(r) = .2
Where I am stuck is how to find the joint probability (or overlap) of dented red gumballs... (Since I am assuming we cannot say that blue and red gumballs are dented equally).
I feel like I should be factoring in P(b) somewhere to find that overlap of r & d, but I'm not sure how...
Any advice on how to find the joint probability of r & d is much appreciated!
(Additionally, the equation to find P(joint r & d)=P(d)*P(r|d)
Let's let $B, R, \text{and } D$ denote the events that we draw a blue, red, and dented gum ball, respectively.
Two events $A$ and $B$ are said to be independent if $\mathsf P(A\cap B)=\mathsf P(A)\cdot\mathsf P(B)$
But $\mathsf P(R\cap B)=0$ and $\mathsf P(R)\cdot\mathsf P(B)=0.2\cdot0.8=0.16$ so $R$ and $B$ are not independent.
In any case, the question asks for the probability of the gum ball being dented and red so we have
$$\mathsf P(D\cap R)=0.2\cdot 0.2=0.04$$