SOLUTION: A high school baseball player has a 0.216 batting average. In one game, he gets 8 at bats. What is the probability he will get at least 4 hits in the game?

Algebra ->  Probability-and-statistics -> SOLUTION: A high school baseball player has a 0.216 batting average. In one game, he gets 8 at bats. What is the probability he will get at least 4 hits in the game?       Log On


   



Question 1207647: A high school baseball player has a 0.216 batting average. In one game, he gets 8 at bats. What is the probability he will get at least 4 hits in the game?

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Answer: 0.07214605 (approximate)
Round it however your teacher instructs.


Explanation

We use the binomial distribution since...
  1. The baseball player gets a hit or they do not. There are two outcomes.
  2. Each trial, or at bat, has the same probability
  3. Each trial is independent.
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 = 0.07214605

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

Think of it like
sum of B(4) through B(8) = 1 - ( sum of B(0) through B(3) )
or,
P(at least 4 hits) = 1 - P(at most 3 hits)

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.

--------------------------------------------------------------------------

A much faster alternative is to use software such as a spreadsheet.
The BinomDist function in Excel and many other spreadsheets is what we want to use.

The command to type in would be =1-BinomDist(3,8,0.216,1)
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 this article
Make sure you use the CDF and not PDF.

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