0.41
There are several different ways to solve this problem. Since there's only 10 coins, you could simply calculate all 2^10 = 1024 possibilities using a spreadsheet and come up with the exact value of 416/1024 = 13/32 = 0.40625, or you can do it manually via reasoning. So:
Let's start with the quarters, and look at the situation of 0, 1, or 2 landing heads up.
0 - The value of all the dimes, nickels, and pennies add up to 39 cents, so Michael can't win.
2 - Got the 50 cents using quarters alone, so this is a 1 in 4 chance. So we have 0.25
1 - This is where it's interesting. And there's a 2 in 4 or 1/2 chance of 1 quarter coming up heads. In this case, we need at least 25 cents worth for the other 8 coins. So let's break down this case.
Let's look at the issue of 0, 1, 2, or 3 dimes.
0 - The nickel and pennies add up to 9, so we don't win.
1 - The nickel and pennies add up to 9, so our total is 19, so we don't win.
3 - We have 30 cents and with the 25 from the quarter we're good to go. This happens 1 out of 8 times. So 1/8 * 1/2 = 1/16 = 0.0625. Adding to the 0.25 gives us 0.3125
2 - We have 20 cents, plus the 25 from the quarter. So we need 5 more cents. We get to this situation 3/8 * 1/2 = 3/16 times.
Nickle
0 = Pennies only add to 4, so can't win.
1 = Gives us the 50 cents we need. So 1/2*3/16 = 3/32 = 0.09375. Adding that to the 0.3125 we already have gives us 0.40625 which exactly matches the exhaustive search of all 1024 possibilities.