In practice, Bayes's formula is very useful when it comes to conditional probabilities. The idea behind the rule is that, when having two events
In this lab, you'll
- learn how to apply Bayes' rule, both in the simple form and the long form
- practice creating two-way tables
Each year, 2000 students apply for undergrad programs at Stanford University. Among the applicants, 20% have an SAT score above 1550, 55% have an SAT score between 1450 and 1550 and 25% have an SAT below 1450.
Obviously, the probability of being admitted grows with the SAT score. People that have an SAT score of 1550 or higher have a probability of 80% of being admitted. For students that have scores between 1450 and 1550 this probability drops to 15%, for students that have an SAT score below 1450 this probability drops to 5%.
- what is the probability of being admitted?
- what is the probability of having an SAT of over 1550 if you were admitted?
what is the probability of being admitted?
According to the decomposition formula:
P_admitted = None
P_admitted # answer : 0.255
what is the probability of having an SAT of over 1550 if you were admitted?
P_1550_given_admitted = None
P_1550_given_admitted # answer: 0.62745
Recall that the probability of being admitted given an SAT of > 1550 is different! So this is a good illustration to show that generally
There is a disease that affects a proportion of
Having the disease is affected by a genetic factor, and we know that the joint probability of you and your father having the disease is equal to
Now our question is: given that your father doesn't have the disease, what is the probability that you won't get the disease? Create a two-way table to solve for this in terms of
P(A) = You have the disease
P(B) = Your dad has the disease
+ | |||
---|---|---|---|
? | ? | ||
? | ? | ? | |
+ | ? | 1 |
Now, let the probability of having the disease be 0.065 and the joint probability of a person and their father having the disease be 0.01, what is the probability that you don't have the disease given that your father doesn't have it?
prob_no_disease = None
prob_no_disease # correct answer: 0.94118
Two boxes are filled with black and white balls. In the first box, there are 3 white balls and 7 black ones, in the second box, there are 6 white balls and 4 black ones. Jenna takes one ball out of box 1 and puts it in the second box without looking at the color. then he takes a ball out of the second box and notices that it's white. Now, it's your turn to compute the probability that Jenna put a white ball from the first box into the second. Use bayes formula to do this! It will help if you define events
Denote event A the event that Jenna put a white ball from the first box to the second.
Denote event B the event that Jenna draws a white ball from box 2.
None
None
None
None
p_a_given_b = None
p_a_given_b # correct answer: 0.33333