The Binomial Probability Distribution
What is a “binomial probability experiment”?
[JK10 page 287]
1. There are 𝑛 repeated independent trials. [The result of one trial does not affect the probability of success on any other trial in the
experiment.]
2. Each trial has two possible outcomes: Success or Failure, Event Happens or Doesn’t Happen, Yes or No, True or False, Right or Wrong.
3. Let 𝑝 = 𝑃(𝑠𝑢𝑐𝑐𝑒𝑠𝑠). Then 𝑞 = 𝑃(𝑛𝑜𝑡 𝑠𝑢𝑐𝑐𝑒𝑠𝑠) = 1 − 𝑝 and 𝑝 + 𝑞 = 1, exactly 1.
4. The random variable, 𝑥, is the count of the number of successful trials. 𝑥 can be any integer value from 0 to 𝑛.
The Binomial Probability Function
𝑛
𝑃(𝑥 ) = ( ) ∙ (𝑝 𝑥 ) ∙ (𝑞 𝑛−𝑥 )
𝑥
This means:
“The Probability of exactly x successes in n trials is computed according to this formula.”
What are the formulas for the Mean and the Standard Deviation?
The Mean number of successes is 𝜇 = 𝑛 ∙ 𝑝
The Standard Deviation is 𝜎 = √𝑛 ∙ 𝑝 ∙ 𝑞
Document1
2/9/2016 3:26 AM
- D.R.S.
Technology examples
8
Example: Rolling a pair of dice 𝑛 = 10 times in hopes of a 7 or an 11, probability 𝑝 = 36. So probability of getting something other than a 7 or
11 is 𝑞 =
28
.
36
Problem statement and By-hand formulas
TI-84 with the binompdf() and binomcdf()
“The probability of there being exactly 𝑥 = 3
successes in the 𝑛 = 10 trials.”
10
( ) ∙ (𝑝3 ) ∙ (𝑞 7 )
3
“The probability of 𝑥 or fewer – that is, 0 to 𝑥,
successes in 𝑛 trials.”
2ND DISTR A: binompdf() to
get the probability at that
one specific 𝑥 value only.
Excel with the =BINOM.DIST()
function (No “.” in Excel 2007).
2ND DISTR B:binomcdf() to get the total probability of all
𝑥 values from 0 to this 𝑥.
𝑥=3
10
∑ ( ) ∙ (𝑝 𝑥 ) ∙ (𝑞10−𝑥 )
𝑥
𝑥=0
(If working by hand, four separate
calculations are needed.)
“The probability of 𝑥 = 3 or more– that is, 𝑥
to 𝑛, successes in 𝑛 trials.”
𝑥=10
10
∑ ( ) ∙ (𝑝 𝑥 ) ∙ (𝑞10−𝑥 )
𝑥
𝑥=3
If working by hand, eight separate
calculations are needed. Or be clever and do
𝑥=2
10
1 − ∑ ( ) ∙ (𝑝 𝑥 ) ∙ (𝑞10−𝑥 )
𝑥
Subtract the cumulative probability of 0 thru 10, minus the
cumulative probability of 0
thru 2.
Or 1 minus the cumulative
probability of 0 thru 2. We
can do this because the
upper bound was the maximum, 10.
𝑥=0
“The probability of at least 𝑎 = 5 but no
more than 𝑏 = 9 successes in 𝑛 trials.”
𝑥=4
10
∑ ( ) ∙ (𝑝 𝑥 ) ∙ (𝑞10−𝑥 )
𝑥
The cumulative probability of
0 thru 9,
Minus the cumulative
probability of 0 thru 4.
𝑥=1
Document1
2/9/2016 3:26 AM
- D.R.S.
Displaying all of the probabilities in a table and a graph
8
Again with 10 rolls of a pair of dice, seeking rolls that total 7 or 11. We have a binomial probability experiment with 𝑛 = 10, and 𝑝 = 36.
TI-84
Define one list with the 𝑥 values
of 0, 1,2, 3, …, 9, 10.
You can inspect the lists if you want, and see all of the probabilities.
Define another list with the
binomial probabilities,
𝑃(0), 𝑃(1), ⋯ , 𝑃(10)
2ND STAT PLOT
We turn this plot on.
We select the bar chart.
WINDOW
Our x values go from 0 to 10, so
we need the x-axis from 0 to 11,
so our bar for x = 10 will be
graphed.
Unnecessary – but we were
curious, so we verified that the
sum fo the probabilities are
exactly 1.
GRAPH
(For “seq(“, use 2ND CATALOG then press the letter S (but not with ALPHA) and scroll down to “seq(“. Do not use the Seq with capital letter S!)
Document1
2/9/2016 3:26 AM
- D.R.S.
Excel
Set up a table of values: For the x values in Column A, you just put the 0 in cell A1. Then hover over the bottom right corner of cell A1 (it’s
called “the fill handle”; see the cursor change to a more compact plus sign), hold down the CTRL key, and drag down through cell A11. Excel
fills the range with consecutive integers for you . This is easier than setting up a formula.
Then in cell B1 type the formula shown.
Click on cell B1, right-click, Copy, then highlight B2 down through B11. Right click, Paste. (Or use the Fill handle, but without the CTRL key).
We did an extra check to make sure that the probabilities sum to exactly 1.
Highlight the column B cells with the probabilities.
Click the ribbon’s “Insert” menu and in the “Charts” section, choose the Column chart. The horizontal axis labels needed to be fixed. The legend
was deleted. Titles and labels should be added, too.
Document1
2/9/2016 3:26 AM
- D.R.S.