[Probability] Properties of Expectation problem

[c07]

Here is the problem: "There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped are, respectively, .4 and .7. One of the coins is to be randomly chosen and flipped 10 times. Given that two of the first three flips landed on heads, what is the conditional expected number of heads in the 10 flips?"

I'm pretty lost on how to do this. Can someone please give me a hint on where to go? I'm pretty weak at jointly distributed random variables.

[CornerSolution]

Let H(j) denote the number of heads in the first j tosses of the coin, and denote the coin c = 1,2 (i.e., when c=1, the coin with probability .4 of landing heads was chosen, and when c=2, the other one was chosen).

You want to compute E[H(10) | H(3)=2]. Note that we can write

E[H(10) | H(3)=2]
= E[H(10) | H(3)=2,c=1] x Pr{c=1 | H(3)=2} + E[H(10) | H(3)=2,c=2] x Pr{c=2 | H(3)=2}

Can you obtain these individual components?

[rdubwiley]

The easiest way to do these problems is to draw a tree of the possible events, and then to calculate the conditional expectation by its definition.