Conditional Probability with R
In this post, we will explore conditional probability and how it relates to Bayes’ Theorem.
Conditional Probability
Conditional probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome. Below is the equation for the conditional probability.
$P(A|B) = \frac{P(A \cap B) }{P(B)}$
where:
P(A ∩ B) is the probability that event A and event B both occur.
P(B) is the probability that event B occurs.
For example:
- Event A is a person completing their projects on time in a year. There is 85% chance that this person will complete their work on time.
- Event B is that this person receiving a raise at the end of the year. Raises are only given to to 45% of those that complete their work on time in a year.
- P(complete work and raise) = P(raise|complete work) P(complete work) = 0.45 * 0.85 = 0.3825
Looks like working hard does pay off. Conditional probability implies there is a relationship between these two events, such as the probability as completing your work on time in a year, and receiving a raise. In other words, conditional probability depends on a previous result. In this case, it is the probability of completing your work on time.
Let’s take a look at a survey given to male and female students and
asking what their favorite past times are. Below a prepared a function
rng_survey() to generate the survey answers.
#helper function for our survey
rng_survey <- function(max, n){
#max is the maximum number of surveys for a participant group (ie. Male)
#n is the number of possible answers
total = 0
while (total != max){
x <- sample(1:max, n, replace = TRUE)
total = sum(x)
}
return(x)
}
#male = rng_survey(150,4)
#female = rng_survey(150,4)
#cat("male answers:", male, "\n")
#cat("female answers:", female, "\n")
#male answers: 7 19 26 98
#female answers: 7 69 40 34
Using the function, we create a data frame with the survey answers.
#create data frame for the survey responses
df <- data.frame(gender=rep(c('Male', 'Female'), each=150),
sport=rep(c('Exercise', 'Cooking', 'Reading', 'Television',
'Exercise', 'Cooking', 'Reading', 'Television'),
times=c(7, 19, 26, 98, 7, 69, 40, 34 )))
df
## gender sport
## 1 Male Exercise
## 2 Male Exercise
## 3 Male Exercise
## 4 Male Exercise
## 5 Male Exercise
## 6 Male Exercise
## 7 Male Exercise
## 8 Male Cooking
## 9 Male Cooking
## 10 Male Cooking
## 11 Male Cooking
## 12 Male Cooking
## 13 Male Cooking
## 14 Male Cooking
## 15 Male Cooking
## 16 Male Cooking
## 17 Male Cooking
## 18 Male Cooking
## 19 Male Cooking
## 20 Male Cooking
## 21 Male Cooking
## 22 Male Cooking
## 23 Male Cooking
## 24 Male Cooking
## 25 Male Cooking
## 26 Male Cooking
## 27 Male Reading
## 28 Male Reading
## 29 Male Reading
## 30 Male Reading
## 31 Male Reading
## 32 Male Reading
## 33 Male Reading
## 34 Male Reading
## 35 Male Reading
## 36 Male Reading
## 37 Male Reading
## 38 Male Reading
## 39 Male Reading
## 40 Male Reading
## 41 Male Reading
## 42 Male Reading
## 43 Male Reading
## 44 Male Reading
## 45 Male Reading
## 46 Male Reading
## 47 Male Reading
## 48 Male Reading
## 49 Male Reading
## 50 Male Reading
## 51 Male Reading
## 52 Male Reading
## 53 Male Television
## 54 Male Television
## 55 Male Television
## 56 Male Television
## 57 Male Television
## 58 Male Television
## 59 Male Television
## 60 Male Television
## 61 Male Television
## 62 Male Television
## 63 Male Television
## 64 Male Television
## 65 Male Television
## 66 Male Television
## 67 Male Television
## 68 Male Television
## 69 Male Television
## 70 Male Television
## 71 Male Television
## 72 Male Television
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## 77 Male Television
## 78 Male Television
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## 80 Male Television
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## 83 Male Television
## 84 Male Television
## 85 Male Television
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## 87 Male Television
## 88 Male Television
## 89 Male Television
## 90 Male Television
## 91 Male Television
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## 93 Male Television
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## 95 Male Television
## 96 Male Television
## 97 Male Television
## 98 Male Television
## 99 Male Television
## 100 Male Television
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## 109 Male Television
## 110 Male Television
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## 115 Male Television
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## 120 Male Television
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## 125 Male Television
## 126 Male Television
## 127 Male Television
## 128 Male Television
## 129 Male Television
## 130 Male Television
## 131 Male Television
## 132 Male Television
## 133 Male Television
## 134 Male Television
## 135 Male Television
## 136 Male Television
## 137 Male Television
## 138 Male Television
## 139 Male Television
## 140 Male Television
## 141 Male Television
## 142 Male Television
## 143 Male Television
## 144 Male Television
## 145 Male Television
## 146 Male Television
## 147 Male Television
## 148 Male Television
## 149 Male Television
## 150 Male Television
## 151 Female Exercise
## 152 Female Exercise
## 153 Female Exercise
## 154 Female Exercise
## 155 Female Exercise
## 156 Female Exercise
## 157 Female Exercise
## 158 Female Cooking
## 159 Female Cooking
## 160 Female Cooking
## 161 Female Cooking
## 162 Female Cooking
## 163 Female Cooking
## 164 Female Cooking
## 165 Female Cooking
## 166 Female Cooking
## 167 Female Cooking
## 168 Female Cooking
## 169 Female Cooking
## 170 Female Cooking
## 171 Female Cooking
## 172 Female Cooking
## 173 Female Cooking
## 174 Female Cooking
## 175 Female Cooking
## 176 Female Cooking
## 177 Female Cooking
## 178 Female Cooking
## 179 Female Cooking
## 180 Female Cooking
## 181 Female Cooking
## 182 Female Cooking
## 183 Female Cooking
## 184 Female Cooking
## 185 Female Cooking
## 186 Female Cooking
## 187 Female Cooking
## 188 Female Cooking
## 189 Female Cooking
## 190 Female Cooking
## 191 Female Cooking
## 192 Female Cooking
## 193 Female Cooking
## 194 Female Cooking
## 195 Female Cooking
## 196 Female Cooking
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## 198 Female Cooking
## 199 Female Cooking
## 200 Female Cooking
## 201 Female Cooking
## 202 Female Cooking
## 203 Female Cooking
## 204 Female Cooking
## 205 Female Cooking
## 206 Female Cooking
## 207 Female Cooking
## 208 Female Cooking
## 209 Female Cooking
## 210 Female Cooking
## 211 Female Cooking
## 212 Female Cooking
## 213 Female Cooking
## 214 Female Cooking
## 215 Female Cooking
## 216 Female Cooking
## 217 Female Cooking
## 218 Female Cooking
## 219 Female Cooking
## 220 Female Cooking
## 221 Female Cooking
## 222 Female Cooking
## 223 Female Cooking
## 224 Female Cooking
## 225 Female Cooking
## 226 Female Cooking
## 227 Female Reading
## 228 Female Reading
## 229 Female Reading
## 230 Female Reading
## 231 Female Reading
## 232 Female Reading
## 233 Female Reading
## 234 Female Reading
## 235 Female Reading
## 236 Female Reading
## 237 Female Reading
## 238 Female Reading
## 239 Female Reading
## 240 Female Reading
## 241 Female Reading
## 242 Female Reading
## 243 Female Reading
## 244 Female Reading
## 245 Female Reading
## 246 Female Reading
## 247 Female Reading
## 248 Female Reading
## 249 Female Reading
## 250 Female Reading
## 251 Female Reading
## 252 Female Reading
## 253 Female Reading
## 254 Female Reading
## 255 Female Reading
## 256 Female Reading
## 257 Female Reading
## 258 Female Reading
## 259 Female Reading
## 260 Female Reading
## 261 Female Reading
## 262 Female Reading
## 263 Female Reading
## 264 Female Reading
## 265 Female Reading
## 266 Female Reading
## 267 Female Television
## 268 Female Television
## 269 Female Television
## 270 Female Television
## 271 Female Television
## 272 Female Television
## 273 Female Television
## 274 Female Television
## 275 Female Television
## 276 Female Television
## 277 Female Television
## 278 Female Television
## 279 Female Television
## 280 Female Television
## 281 Female Television
## 282 Female Television
## 283 Female Television
## 284 Female Television
## 285 Female Television
## 286 Female Television
## 287 Female Television
## 288 Female Television
## 289 Female Television
## 290 Female Television
## 291 Female Television
## 292 Female Television
## 293 Female Television
## 294 Female Television
## 295 Female Television
## 296 Female Television
## 297 Female Television
## 298 Female Television
## 299 Female Television
## 300 Female Television
We convert the data frame to a table.
#create two-way table from data frame
survey_data <- addmargins(table(df$gender, df$sport))
survey_data
##
## Cooking Exercise Reading Television Sum
## Female 69 7 40 34 150
## Male 19 7 26 98 150
## Sum 88 14 66 132 300
We can extract information from our table by calling a row and a column. For instance, let’s ask for the number of males that prefer cooking.
survey_data['Male', 'Cooking']
## [1] 19
Now we can ask the probability of being male given that they prefer
cooking. We know that the probability of being male is 0.5. We can
calculate the rest from the table.
P_male = 0.5
P_cooking_male = survey_data['Male', 'Cooking'] / survey_data['Male', 'Sum'] #probability of only males that prefer cooking
P_male_P_cooking_male = P_male * P_cooking_male
P_cooking = survey_data['Sum', 'Cooking'] / survey_data['Sum', 'Sum'] #probability of male and female that prefers cooking
P_male_cooking = P_male_P_cooking_male / P_cooking
P_male_cooking #probability of being male given they prefer cooking
## [1] 0.2159091
Alternatively, we can use the table to easily answer the same problem.
survey_data['Male', 'Cooking'] / survey_data['Sum', 'Cooking']
## [1] 0.2159091
Next, we can ask the probability of being female given that they prefer reading.
survey_data['Female', 'Reading'] / survey_data['Sum', 'Reading']
## [1] 0.6060606
Bayes’ Theorem
Suppose that we have the same survey but male and female was not recorded by accident. How could we solve the probability of being male given that they prefer cooking? Let us assume from a previous survey we knew that 12.67% of males prefer to cook or is the probability of preferring to cook given that they are male. We can use something called Bayes’ Theorem,
$P(A|B) = \frac{P(B|A) P(A) }{P(B)}$.
Bayes’ Theorem describes the probability of an event based on prior knowledge of conditions that may be related to the event.
Knowing that the equation for conditional probability is
$P(A|B) = \frac{P(A \cap B) }{P(B)}$
then
P(A ∩ B) = P(A|B)P(B)
and
P(A ∩ B) = P(B|A)P(A).
We can solve for P(A ∩ B) with substitution to yield
$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$.
#view table
survey_data
##
## Cooking Exercise Reading Television Sum
## Female 69 7 40 34 150
## Male 19 7 26 98 150
## Sum 88 14 66 132 300
P_male = 0.5
P_cooking_male = 0.1267 #probability of prefering to cook given that they are male
P_cooking = survey_data['Sum', 'Cooking'] / survey_data['Sum', 'Sum'] #probability of male and female that prefers cooking
P_male_cooking = (P_cooking_male * P_male) / P_cooking
P_male_cooking #probability of being male given they prefer cooking
## [1] 0.2159659
Let’s try to solve for the reverse, the probability of preferring to cook given that they are male, using Bayes’ Theorem.
P_male_cooking * P_cooking / P_male
## [1] 0.1267
Is there a way to solve for the probability of preferring to cook given that they are female using Bayes’ Theorem?
P_female = 1 - P_male #We have a binary choice and the probabilities sum up to 1
P_female_cooking = 1 - P_male_cooking
P_female_cooking * P_cooking / P_female
## [1] 0.4599667
Looking back at the table, we can use Bayes’ Theorem to solve this problem.
survey_data['Female', 'Cooking'] / survey_data['Female', 'Sum']
## [1] 0.46