probability of getting decoder rings in cereal boxes

[tsh44]

Hello, I having trouble with this problem.

One out of every five boxes of cereal has a secret message decoder ring. You buy 5 boxes hoping to get at least two rings. What are your chances.

The answer should be 821/3125.

Thanks for any help.

 

[soroban]

Re: probability

Hello, tsh44!

One out of every five boxes of cereal has a secret message decoder ring.
You buy 5 boxes hoping to get at least two rings. What are your chances?

The answer should be 821/3125.

You're expected to be familiar with Binomial Probability (or Independent Trials)
$\displaystyle \;\;$and its formula.
We are given: $\displaystyle \,P(\text{ring})\,=\,\frac{1}{5},\;\;P(\text{no ring})\,=\,\frac{4}{5}$

The opposite of "at least two" is "zero or one".

$\displaystyle P(\text{0 rings})\;=\;C(5,0)\cdot\left(\frac{1}{5}\right)^o\left(\frac{4}{5}\right)^5\;=\;\frac{1024}{3125}$

$\displaystyle P(\text{1 ring})\;=\;C(5,1)\cdot\left(\frac{1}{5}\right)^1\left(\frac{4}{5}\right)^4\;=\;\frac{1280}{3125}$

Hence: $\displaystyle \,P(\text{0 or 1 ring})\;=\;\frac{1024}{3125}\,+\,\frac{1280}{3125}\;=\;\frac{2304}{3125}$


Therefore: $\displaystyle \,P(\text{2 or more rings})\;=\;1\,-\,\frac{2304}{3125}\;=\;\frac{821}{3125}$

 

[tsh44]

thank you