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Let's say I have a coin weighted toward heads – 65% heads. What's the chance I will get at least one head if I have to flip the coin 5 times?

Basic probability and intuition tells me $\frac{0.65}5$.

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The answer is not $\frac{65}5$, $\frac{0.65}5$ or similar things to those. To get the correct answer: there is a 0.35 chance of getting tails, so the probability that you get five tails and no heads is $0.35^5$. The complement probability, that of getting some head in five tosses, is the complement of this, or $1-0.35^5=0.9947$ – i.e. around 99.5%.

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You need to check your intuition - how does increasing the number of throws decrease the probability of at least one head (you divide by $5$)?

Another answer has given you the right calculation, but here are some things to think about so that you start noticing the right things here:

  1. As the number of flips increases, the chance of at least one head does not decrease (and in fact increases)

  2. It is always possible, though increasingly unlikely as the number of flips increases, that all the flips will be tails, so the probability of at least one head is never equal to $1$

So if you get an answer, or an intuition, or a result of applying "basic probability", which does not conform to these observations, then something has done wrong with your thinking. That should be a big hint to get you thinking in the right way about the problem.

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