Easy Coin Problems Made Easy

This side applies to both Math 1001 Quantitative Reasoning and Math 2411 Introduction to Statistics.

Every Classical Probability Problem Comes Down to This Fraction: / P(event occurs)= / count how many ways for the event to occur
count how many outcomes in the sample space, total

Count How Many Ways for All Heads / No Tails

/

Count Outcomes in the Sample Space

Only one way for it to happen! Numerator = 1
All Heads in Two Coin Tosses: only HH
All Heads in Three Coin Tosses: only HHH
All Heads in Four Coin Tosses: only HHHH
All Heads in Five Coin Tosses: only HHHHH
Ten Tosses? only HHHHHHHHHH / There are 2 possibilities for the first toss: H, T
There are 2 possibilities for the second toss: H, T
2 matched with 2 more = 2 ∙ 2 = 4 possibilities
There are 2 possibilities for the third toss: H, T
2 matched with 2 matched with 2 = 2 ∙ 2 ∙ 2 = 8
There are 2 possibilities for the fourth toss: H, T
2 ∙ 2 ∙ 2 ∙ 2 = 16
There are 2 possibilities for the fifth toss: H, T
2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 32
Etc., etc. Each toss grows two new tree branches.

How Many Ways for One Head &the rest Tails

How many ways? Same as number of tosses. Why?
If 2 tosses, 2 ways: / HT or TH
If 3 tosses, 3 ways: / HTT or THT or TTH
If 4 tosses, 4 ways: / HTTT or THTT or TTHT or TTTH / The total number of outcomes is
2how many tosses
If 10 tosses, then there are outcomes.
Because the one H can appear in the 1st, in the 2nd, in the 3rd, etc. tosses, up to the number of tosses.
If 10 tosses, 10 ways / HTTTTTTTTT etc. …TTTTTTTTTH

What if it’s a Tails problem? Like “all Tails” or “1 Tail”? ►Same methods, just a different perspective.

What if it’s like “9 coin tosses, 8 tails”? ►Same as “9 tosses, 1 head”.

What if it’s like “2 heads in 9 tosses” or “7 heads in 9 tosses”? ►See other side.

Harder Coin Problems Made Easy

This side is generally needed by Statistics class only.

Every Classical Probability Problem Comes Down to This Fraction: / P(event occurs)= / count how many ways for the event to occur
count how many outcomes in the sample space, total

Count how many ways to Get “r” Heads in “n” Tosses

/

Count Outcomes in the Sample Space

Think in the word pattern “ things taken at a time” / As described on the front side of these notes:
The total number of outcomes in the sample space is
2how many tosses
The order of the Heads does not matter.
  • HHHTT is 3 heads, and so is THTHH 3 heads, too.
  • Plus several other ways to get 3 heads in 5 tosses.

Therefore it is a COMBINATION.
On the TI-84 Calculator,
  • Type the number of tosses first.
  • Then MATH and go to the, PRB submenu, 3:nCr.
  • Then type the number and press ENTER.
On the TI-30 XS Multiview Calculator,
  • Type the number of tosses first.
  • Press PRB button and pick 2:nCr.
  • Then type the number and press ENTER.

Example: If 10 coin tosses
and we are interested in the event “exactly 6 Heads”,
then the calculation is 10 nCr 6 = 210 outcomes for this event. / Example: If 10 coin tosses,
then there are outcomes
in the sample space.
Example – Putting it all together – the probability of getting exactly 6 heads in 10 coin tosses is:

10_Coin_Toss.docx 10/19/2016 9:11 AM D.R.S.