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A coin is such that after every toss the probability of same side coming again increases by $50\%$ from the initial value. If initially the probability of head and tail are the same, what is the expected number of tosses until we get a head?

$E_{\text{tosses}} = 0.5 \times 1 + 0.5 \times(1+ E_{HaT})$

$\implies E_{\text{tosses}} = 1 + 0.5 E_{HaT}$

$E_{HaT} = 0.25 \times 1 + 0.75 \times (1 + E_{HaT})$

$\implies E_{HaT} = 1 + 0.75 E_{HaT}$

$\implies E_{HaT} = \frac{1}{0.25} = 4$

$\therefore E_{\text{tosses}} = 1 + 0.5 \times 4 = 3$

So, we must toss the coin at least $3$ times to expect a Head.
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