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# Probability Problem

A man tosses a coin $$10$$ times, scoring $$1$$ point for each head and $$2$$ points for each tail. Let $$P(k)$$ be the probability of scoring at least $$k$$ points.

Find the largest value of $$k$$ such that $$P(k)>\frac{1}{2}$$

Note by Krishna Jha
3 years, 10 months ago

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We can make use of a generating function to get the number of ways to score points:

$$G(z) = (z^1+z^2)^{10}$$

The coefficient of each $$z^i$$ for i=10..20 is the number of ways of scoring 'i' points.

We can see that the largest k occurs in the middle of the expansion, which is k=15

$$P(k=15) = \dfrac{\sum_{i=5}^{10}\binom{10}{i}}{2^{10}}=\frac{638}{1024}=0.623046875$$ · 3 years, 10 months ago

20 · 1 year, 11 months ago