John has $\ 6 $ coins. He performs a test which involves tossing all coins simultaneously. He repeats this experiment until all coins shows up as heads. What is the probability that it takes him at least $\ 12 $ tosses to achieve this feat?
A probability question (outcomes)
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0Hint: Think of the geometric distribution. – 2012-12-06
2 Answers
The probability to fail at any toss is $p=1-\frac1{2^6}$ (why?). The probability of failing $11$ times in a row is $p^{11}$.
The probability of having all $6$ coins come up heads simultaneously on any given throw is $(1/2)^6 = 1/64$
The Geometric Distribution is used to determine the number of Bernoulli trials needed to get $1$ success. What we are seeking here is the probability that a series of Bernoulli trials with probability $1/64$ takes longer than $12$ trials to achieve a success.
The cumulative distribution function for the Geometric distribution is $1 - (1-P)^k$, where $P$ is the probability of success and $k$ is the number of trials. The table below is computed by substituting $1/64$ for $P$ and the value in the first column for $k$.
$k$ $P(n\le k)$
$1$ $0.0156$
$2$ $0.031$
$3$ $0.046$
$4$ $0.061$
$5$ $0.075$
$6$ $0.090$
$7$ $0.104$
$8$ $0.118$
$9$ $0.132$
$10$ $0.145$
$11$ $0.159$
The bottom line tells us that there is approximately a $15.9\%$ chance that we will have a success by the $12$th trial. Therefore there is approximately and $84.1\%$ chance that it will take at least $12$ tries before getting all $6$ coins to show heads.