Question 1207647
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Answer:  <font color=red>0.07214605</font> (approximate)
Round it however your teacher instructs.



Explanation


We use the binomial distribution since...<ol><li>The baseball player gets a hit or they do not. There are two outcomes. </li><li>Each trial, or at bat, has the same probability</li><li>Each trial is independent. </li></ol>The three criteria for a binomial have been met.


The binomial probability formula is
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
where,
n = sample size = number of attempts or at bats = 8
p = batting average = probability of a hit = 0.216
x = number of hits
x spans from x = 0 to x = 8.
i.e. x is from the set {0,1,2,3,4,5,6,7,8}


The nCx refers to the nCr combination formula. 
Some textbooks call it the "choose" formula. 
These values can be computed with the nCr formula or by use of Pascal's Triangle.


Let's compute the binomial probability that the player gets x = 0 hits
B(x) = (nCx)*(p^x)*(1-p)^(n-x)
B(x) = (8Cx)*(0.216^x)*(1-0.216)^(8-x)
B(x) = (8Cx)*(0.216^x)*(0.784)^(8-x)
B(0) = (8C0)*(0.216^0)*(0.784)^(8-0)
B(0) = 1*(0.216^0)*(0.784)^(8-0)
B(0) = 0.14273435
There is roughly a 14.27% chance that the batter doesn't get any hits in the 8 at bats.


Follow similar steps for x = 1 through x = 3 to find:
B(1) = 0.31459816
B(2) = 0.30336251
B(3) = 0.16715893


Then adding up those results gives
B(0) + B(1) + B(2) + B(3) 
= 0.14273435 + 0.31459816 + 0.30336251 + 0.16715893
= 0.92785395
This is the probability that the batter gets at most 3 hits, i.e. 3 or less hits.


Subtract this from 1 to get the final answer.
1 - 0.92785395 = <font color=red>0.07214605</font>


Why are we subtracting from 1? 
Because of this scratch work
( <font color=blue>B(0) + B(1) + B(2) + B(3)</font> ) + ( <font color=red>B(4) + B(5) + B(6) + B(7) + B(8)</font> ) = 1
which solves to
<font color=red>B(4) + B(5) + B(6) + B(7) + B(8)</font> = 1 - ( <font color=blue>B(0) + B(1) + B(2) + B(3)</font> )


Think of it like 
<font color=red> sum of B(4) through B(8) </font> = 1 - ( <font color=blue> sum of B(0) through B(3) </font>)
or,
<font color=red> P(at least 4 hits) </font> = 1 - <font color=blue>P(at most 3 hits) </font>


Or alternatively you can compute values B(4) through B(8) and add them up to get the answer more directly. 
There's not much of a trade-off between both options since we're adding up the same number of terms more or less.


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A much faster alternative is to use software such as a spreadsheet.
The <a href="https://support.microsoft.com/en-us/office/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c">BinomDist</a> function in Excel and many other spreadsheets is what we want to use.


The command to type in would be <font color=red>=1-BinomDist(3,8,0.216,1)</font>
The BinomDist(3,8,0.216,1) portion computes the sum from B(0) up to B(3) including both endpoints. Then we subtract from 1 as explained earlier.
The equal sign up front is to tell the spreadsheet software to compute the formula rather than display as plaintext.
Refer to the spreadsheet help manual, or the link I posted, for more information how the BinomDist function works.


If you have access to a TI83 calculator or similar, then check out <a href="https://www.statology.org/binomial-probabilities-ti-84-calculator/">this article</a>
Make sure you use the CDF and not PDF. 


Various free online binomial distribution calculators can achieve this same goal.
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