Biased Coins

You have six different coins. When tossed, 3 of the coins are fair and give heads 50% of the time, 2 of the coins are slightly biased and give heads 75% of the time and 1 coin is completely biased and gives heads 100% of the time.

You choose one of the coins completely at random and toss the coin twice.

If both of the tosses are heads, the probability that you chose a fair coin can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are positive, co-prime integers. Calculate the value of \(a + b\).

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