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The question is:

Probability of scoring $1$ out of $3$ shots when you have 80% throw rate.

I solved this problem in the inverse way:

P(At least one basket) = $1$ - P(No basket) = $1$ - ($.20 \times .20 \times .20) = 0.992 ~ or ~99.2 \% $

I wanted to know how would I go about solving this the other (which I realize is not the best way)..

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Which do you mean, "at least one" or "exactly one"? – Robert Israel Aug 17 '12 at 5:39
If I had to put some money on the exact meaning of Probability of scoring 1 out of 3 shots, I would bet on exactly one hit. Then the numerical answer is 9.6%. – Did Aug 17 '12 at 7:52
I agree that this is confusing, but technically, if you score twice, you have scored once. Also, the questioner supports this via his/her calculations. She also mentions P(At least one basket), which isn't very ambiguous. – C. Williamson Aug 17 '12 at 17:40
@C.Williamson: The text is 1 out of 3, the rest is an interpretation by the OP. – Did Aug 17 '12 at 22:43
@RobertIsrael its actually atleast – MistyD Aug 18 '12 at 13:37

The way you do this is $3\choose{1}$(.8*.2*.2)+$3\choose{2}$(.8*.8*.2)+$3\choose{3}$$(.8^3)$ This equals 0.992. There are $3\choose{1}$ ways to get one shot, $3\choose{2}$ ways to get two and $3\choose{3}$ ways to get all of them. Note that I could have just as easily counted misses rather than successful shots, which is okay, since binomial coefficients satisfy $n\choose{k}$=$n\choose{n-k}$.

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A way which nobody has mentioned is this:

The probability is: getting the first throw + missing the first throw but getting the second + missing the first 2 but getting the third.


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Assuming independence between the results of every two shots, then the scoring variable $K$ out of $n$ shots with success rate of $p$ is (we write $K$~$B(n,p)$)

$$ Pr(K=k)=\binom{n}{k}p^k(1-p)^{n-k} $$

($K=2$ means two shots of success, and $Pr()$ means probability.)

So in your case: $$\begin{align} Pr(K>0)&=Pr(K=3)+Pr(K=2)+Pr(K=1) \\ &={3\choose 3}0.8^3 + {3\choose 2}0.8^2 0.2^1 + {3\choose 1}0.8^1 0.2^2 \\ &=0.512+0.384+0.096 \\ &=0.992 \\ &=1-Pr(K=0) \end{align}$$

So, actually $1-Pr(K=0)$ would be the best way to go for -- easy calculation.

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