11.2: Conditional Probability (Section 10.4)
- Page ID
- 8782
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
Let’s determine the conditional probability of someone being unhealthy, given that they are over 70 years of age, using the NHANES dataset. Let’s create a new data frame that
healthDataFrame <-
NHANES %>%
mutate(
Over70 = Age > 70,
Unhealthy = DaysPhysHlthBad > 0
) %>%
dplyr::select(Unhealthy, Over70) %>%
drop_na()
glimpse(healthDataFrame)
## Observations: 4,891
## Variables: 2
## $ Unhealthy <lgl> FALSE, FALSE, FALSE, TRUE, FALSE, TRUE,…
## $ Over70 <lgl> FALSE, FALSE, FALSE, FALSE, FALSE, FALS…
First, what’s the probability of being over 70?
pOver70 <-
healthDataFrame %>%
summarise(pOver70 = mean(Over70)) %>%
pull()
# to obtain the specific value, we need to extract it from the data frame
pOver70
## [1] 0.11
Second, what’s the probability of being unhealthy?
pUnhealthy <-
healthDataFrame %>%
summarise(pUnhealthy = mean(Unhealthy)) %>%
pull()
pUnhealthy
## [1] 0.36
What’s the probability for each combination of unhealthy/healthly and over 70/ not? We can create a new variable that finds the joint probability by multiplying the two individual binary variables together; since anything times zero is zero, this will only have the value 1 for any case where both are true.
pBoth <- healthDataFrame %>%
mutate(
both = Unhealthy*Over70
) %>%
summarise(
pBoth = mean(both)) %>%
pull()
pBoth
## [1] 0.043
Finally, what’s the probability of someone being unhealthy, given that they are over 70 years of age?
pUnhealthyGivenOver70 <-
healthDataFrame %>%
filter(Over70 == TRUE) %>% # limit to Over70
summarise(pUnhealthy = mean(Unhealthy)) %>%
pull()
pUnhealthyGivenOver70
## [1] 0.38
# compute the opposite:
# what the probability of being over 70 given that
# one is unhealthy?
pOver70givenUnhealthy <-
healthDataFrame %>%
filter(Unhealthy == TRUE) %>% # limit to Unhealthy
summarise(pOver70 = mean(Over70)) %>%
pull()
pOver70givenUnhealthy
## [1] 0.12