# If you know that it rains during ten days out

If you know that it rains during ten days out of every thirty days in a particular city. Using the binomial distribution, what is the probability that no rain will fall during a given week?

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Obiajulu
Step 1
In statistics theory, the Binomial distribution is the discrete probability distribution and it is a two-parameter distribution. The probability of success for each binomial trial is assumed to be independent and same.
Step 2
Given Information:
Let the random variable X denotes the number of days the rain will fall.
The number of days observed (n) is 7.
It is known that it rains during ten days out of every thirty days in a particular city. Hence, the probability of success (p) that is proportion of days it will rain is $$\displaystyle{\frac{{{10}}}{{{30}}}}$$ which is same as $$\displaystyle{\frac{{{1}}}{{{3}}}}$$.
Since, each day is independent of the other and probability of success is same for each day. Hence, assume,
$$\displaystyle{X}\sim{B}\in{\left({7},{\frac{{{1}}}{{{3}}}}\right)}$$
The mass function of X can be obtained from statistical tables:
$$\displaystyle{P}{\left({X}={x}\right)}=^{{{7}}}{C}_{{{x}}}{\left({\frac{{{1}}}{{{3}}}}\right)}^{{{x}}}{\left({1}-{\frac{{{1}}}{{{3}}}}\right)}^{{{7}-{x}}};{x}={0},{1},{2},\ldots\ldots,{7}$$
Step 3
Using the binomial distribution, the probability that no rain will fall during a given week can be obtained as:
$$\displaystyle{P}{\left({X}={0}\right)}=^{{{7}}}{C}_{{{0}}}{\left({\frac{{{1}}}{{{3}}}}\right)}^{{{0}}}{\left({1}-{\frac{{{1}}}{{{3}}}}\right)}^{{{7}-{0}}}$$
$$\displaystyle={\left({1}\right)}{\left({1}\right)}{\left({\frac{{{2}}}{{{3}}}}\right)}^{{{7}}}$$
$$\displaystyle={0.059}$$
$$\displaystyle={0.059}$$
Therefore, using the binomial distribution, the probability that no rain will fall during a given week is 0.059.
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