Notes: If you throw a dice six times ,what's the chance that you get a six on exactly one of the throws ?

Friday, July 13, 2012

If you throw a dice six times ,what's the chance that you get a six on exactly one of the throws ?

We are asked to find the probability of rolling exactly 1 six when rolling a standard die 6 times.

This is a binomial experiment: there are a fixed number of trials (6), there are only two outcomes (a 6 or not a 6), the outcomes are independent and the probability remains the same for each trial.

The formula is $\displaystyle{\left(\matrix{{n}\\{x}}\right)}{{p}}^{{x}}{{q}}^{{{n}-{x}}}$ where $\displaystyle{\left(\matrix{{n}\\{x}}\right)}$ is the number of combinations of n objects taken x at a time, p is the probability of a success, and q is the complement of p (q=1-p). Here n=6, p=1/6, q=5/6, and x=1:

$\displaystyle{\left(\matrix{{6}\\{1}}\right)}{{\left(\frac{{1}}{{6}}\right)}}^{{1}}{{\left(\frac{{5}}{{6}}\right)}}^{{5}}={{\left(\frac{{5}}{{6}}\right)}}^{{5}}=\frac{{3125}}{{7776}}\approx{.402}$

Thus P(x=1) is 3125/7776 or about .402

We could also get this from first principles:

There are 6 rolls. If only 1 is a 6, then there is 1 six with probability 1/6, and 5 other numbers each with probability 5/6. The 6 can occur in any of the six places so we have $\displaystyle{6}\cdot{{\left(\frac{{1}}{{6}}\right)}}^{{1}}\cdot{{\left(\frac{{5}}{{6}}\right)}}^{{5}}$ as above.

Notes

Friday, July 13, 2012

If you throw a dice six times ,what's the chance that you get a six on exactly one of the throws ?

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