Probability of choosing an item dependent on a previous choice A box contains $2$ new and one used radio. the radios are numbered $1$,$2$,$3$. A second box contains only one radio. A radio from the first box was chosen and placed in the second. A radio was chosen then from the second box. What is the probability that the radio chosen was new?
I dont know how to go about this problem. The prob of picking a new one in the first box is $\frac{2}{3}$ but then picking a new radio frm the second would be $100\%$. So i need to start off with the prob of not picking a new one? Please start me off.
 A: To start off, use the law of total probability. I'll give you an analogous problem. Let's say the chance of going to work by bicycle is 1/4, $$P(B) = 1/4$$, and the chance of going to work by car is 3/4, $$P(\neg B) = 3/4$$. The chance of being late if you go to work by bicycle is 1/4, $$P(L|B)=1/4$$ and the chance of being late if you go to work by car is 1/2, $$P(L|\neg B) = 1/2$$, noting that the symbol $\neg$ means "not".
We can use the theorem of total probability to work out the chance of being late given the conditional probabilities of being late over a set of mutually exclusive events (going by bike or not going by bike).
So $$P(L) = P(L|B)P(B) + P(L|\neg B)P(\neg B)$$
$$P(L) = 1/4 \times 1/4  + 1/2 \times 3/4 = 7/16$$
In your case, your mutually exclusive events (corresponding to B) are the events "pick new" or "not pick new" on your first pick. And the event you want to evaluate (corresponding to L) is "pick new from second". Does this help?
A: A Start: The event "New out of the second box" can happen in two mutually exclusive ways: (i) We transferred a new radio from the first box, and got a new when we chose from the second or (ii) We transferred an old from the first box, and got a new when we chose from the second. 
If it will help, draw a tree diagram that describes the two paths by which we could end up with a new radio.
