Question

There are 12 red checkers and 3 black checkers in a bag. Checkers are selected one at a time, with replacement. Each time, the color of the checker is recorded. Find the probability of selecting a red checker exactly 7 times in 10 selections. Show your work.

Answer 1

I will use the binomial distrubion to find the probability.

$\displaystyle P(A)=\binom{n}{k}p^k(1-p)^{n-k}$

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A - selecting a red checker exactly 7 times in 10 selections

$n=7\\ k=10$

$\displaystyle P(A)=\binom{10}{7}\cdot\left(\dfrac{12}{15}\right)^7\cdot\left(1-\dfrac{12}{15}\right)^{10-7}\\\\ P(A)=\dfrac{10!}{7!3!}\cdot\left(\dfrac{4}{5}\right)^7\cdot\left(1-\dfrac{4}{5}\right)^3\\\\ P(A)=\dfrac{8\cdot9\cdot10}{2\cdot3}\cdot\dfrac{16384}{78125}\cdot\left(\dfrac{1}{5}\right)^3\\\\ P(A)=120\cdot\dfrac{16384}{78125}\cdot\dfrac{1}{125}\\\\ P(A)=\dfrac{393216}{1953125}\approx0.2=20\%$