Probability Problem

This appeared in K.V.P.Y. 2012... Please help.

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 $P(k)>\frac{1}{2}$

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

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

$P(k=15) = \dfrac{\sum_{i=5}^{10}\binom{10}{i}}{2^{10}}=\frac{638}{1024}=0.623046875$

gopinath no - 7 years, 8 months ago

pls check for the soln on the following link www.careerpoint.ac.in

sumedh bang - 7 years, 8 months ago

Akhil Kumar - 5 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$