Laws of Probability and
Chi Square
Probability & Genetics
AP Biology
Probability & Genetics
 Calculating probability of making a
specific gamete is just like calculating
the probability in flipping a coin
 probability of tossing heads? 50%
 probability making a P gamete…
 Outcome of 1 toss has no impact on the
outcome of the next toss
 probability of tossing heads each time? 50%
 probability making a P gamete each time?
AP Biology
Rule of Addition
 Chance that an event can occur
2 or more different ways

SUM of the separate probabilities
 Use for heterozygous possibilities
Two ways to be heterozygous: Pp or pP
 Key word is “or”.

 Ex: Probability of getting 2 or a 6 on the roll of a
die. 1/6 + 1/6 = 2/6 = 1/3
 Ex: Probability of having offspring with dominant
phenotype? PP or Pp or pP ¼+ ¼ + ¼ = ¾
AP Biology
Rule of multiplication
 Chance that 2 or more independent
events will occur together
probability that 2 coins tossed at the
same time will land heads up
 probability of pp or PP offspring

 Ex: Probability of getting a head and a
tail with two different coins.
½ x ½ = 1/4
 Key word is “and”

AP Biology
Calculating
Probability of
Pp x Pp
½ x ½ = ¼ = PP
½ x ½ = ¼ = pp
 What about
Pp?
AP Biology
Calculating Dihybrid Probability
 Rule of multiplication application with
Dihybrid crosses:
heterozygous parents — YyRr
 probability of producing yyrr?
 probability of producing y gamete = 1/2
 probability of producing r gamete = 1/2
 probability of producing yr gamete

 = 1/2 x 1/2 = 1/4

probability of producing a yyrr offspring
 = 1/4 x 1/4 = 1/16
AP Biology
What is Chi-Squared?
 In genetics, you can predict genotypes


based on probability (expected results)
Chi-squared is a form of statistical
analysis used to compare the actual
results (observed) with the expected
results
NOTE: 2 is the name of the whole
variable – you will never take the
square root of it or solve for 
AP Biology
Chi-squared
 If the expected and observed (actual)
values are the same then the 2 = 0
 If the 2 value is 0 or is small then the
data fits your hypothesis (the expected
values) well.
 By calculating the 2 value you
determine if there is a statistically
significant difference between the
expected and actual values.
AP Biology
Step 1: Calculating 2
 First, determine what your expected and



observed values are.
Observed (Actual) values: That should
be something you get from data– usually
no calculations 
Expected values: based on probability
Suggestion: make a table with the
expected and actual values
AP Biology
Step 1: Example
 Observed (actual) values: Suppose you


have 90 tongue rollers and 10
nonrollers
Expected: Suppose the parent
genotypes were both Rr  using a
punnett square, you would expect 75%
tongue rollers, 25% nonrollers
This translates to 75 tongue rollers, 25
nonrollers (since the population you
are dealing with is 100 individuals)
AP Biology
Step 1: Example
 Table should look like this:
Expected
Tongue rollers
75
Observed
(Actual)
90
Nonrollers
25
10
AP Biology
Step 2: Calculating 2
 Use the formula to calculated 2
 For each different category (genotype

or phenotype calculate
(observed – expected)2 / expected
Add up all of these values to determine
2
AP Biology
Step 2: Calculating 2
AP Biology
Step 2: Example
 Using the data from before:
 Tongue rollers


(90 – 75)2 / 75 = 3
Nonrollers
(10 – 25)2 / 25 = 9
2 = 3 + 9 = 12
AP Biology
Step 3: Determining Degrees of
Freedom
 Degrees of freedom = # of categories –


1
Ex. For the example problem, there
were two categories (tongue rollers and
nonrollers)  degrees of freedom = 2
–1
Degrees of freedom = 1
AP Biology
Step 4: Critical Value
 Using the degrees of freedom,

determine the critical value using the
provided table
Df = 1  Critical value = 3.84
AP Biology
Step 5: Conclusion
 If 2 > critical value…
there is a statistically significant
difference between the actual and
expected values.
 If 2 < critical value…
there is a NOT statistically significant
difference between the actual and
expected values.
AP Biology
Step 5: Example
 2 = 12 > 3.84
There is a statistically significant
difference between the observed and
expected population
AP Biology
 Bozeman Chi-squared test video
 http://www.youtube.com/watch?v=WXP
BoFDqNVk&edufilter=vBrBKiVMlaMnBr
dX3oXR-Q&safe=active
 Fill in video questions
AP Biology
Animal Behavior Chi Square
Wet
Dry
Observed value
8.9
1.1
Expected value
5
5
Chi Square = Σ (O – E)2 / E
(8.9 – 5)2 / 5
+
(1.1-5)2 / 5
15.21 / 5
+
15.21 / 5
30.42/ 5
AP Biology
=
6.084
2 variable – wet and dry so
1 degree of freedom
6.084 is higher than 3.841 so must reject null hypothesis.
Something influenced the Pill bugs.
AP Biology
 In 2002 The distribution for Skittles is:
Green: 19.7%, Yellow: 19.5%,
Orange: 20.2%, Red: 20%,
Purple: 20.6%.
 Color distribution for M&Ms
Brown15% Yellow12% Orange20%
Red 13% Green 16% Blue 24%
AP Biology