Flipping Biased Coins

After a break, we are back with coin-flipping games. Here is the first – A biased coin produces heads 70% of the time. You toss the coin twice. If both tosses have the same outcome, what is the probability that both tosses are tails?

Let’s apply the general form of Bayes’ equation straightaway.

$P(TT|Same) = \frac{P(Same|TT) * P(TT)}{P(Same|TT) * P(TT) + P(Same|HT) * P(HT) + P(Same|TH) * P(TH) + P(Same|HH) * P(HH)} \\ \\ = \frac{1 * 0.3*0.3}{1 * 0.3*0.3 + 0 + 0 + 1 * 0.7*0.7} = \frac{0.09}{0.58} = 0.155 \\ \\$

Second one: there are two kinds of coins in a box in equal numbers – fair coins and the biased coins of the previous type (70% heads). You randomly select one and flip it twice. If it lands tails on both occasions, What is the probability that the coin is biased?

$P(Biased|TT) = \frac{P(TT|Biased) * P(Biased)}{P(TT|Biased) * P(Biased) + P(TT|NOT-Biased) * P(NOT-Biased)} \\ \\ = \frac{0.3*0.3*0.5}{0.3*0.3*0.5+ 0.5*0.5*0.5} = \frac{0.045}{0.17} = 0.265 \\ \\$

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