PA-4 / Lesson 5
Problem 1.
In the first urn, there are 10 balls, 8 of which are white. In the second urn, there are 20 balls, 4 of which are white. From each urn, one ball is randomly drawn, and then one of these two balls is randomly chosen. What is the probability that the chosen ball is white?
Problem 2.
In a class there are 25 children. Two are chosen for duty. The probability both are boys is 3/25. How many girls are there in the class?
Problem 3.
Two people throw a fair die, one by one, until one of them rolls a six. What are their chances of winning this game?
Problem 4.
There are two coins: one is fair, another has Heads on both sides. One of the coins is picked at random, and then tossed 3 times. You do not see which coin has been picked, but you are told that it was three heads. What is the probability that the 4th flip will also be heads?
Problem 5.
There is a random number generator which writes down a new number every minute. Each of those numbers is an integer between 0 and 2022. Let $P_k$ denote the probability that at some points the sum of all the numbers written down is equal to $k$. Which of the two numbers is larger: $P_{2023}$ or $P_{2022}$?
One example of conditional probability that appears counter-intuitive when first encountered is the following situation. In practice, it arises frequently. Consider one individual chosen at random from 50 men and 50 women applicants to a particular College. Figures on the 100 applicants are given in the following table indicating whether they were educated at a state school or at an independent school and whether they were admitted or rejected.
Note that overall the probability that an applicant is admitted is 0.53, but conditional on the candidate being from an independent school the probability is 0.56 while conditional on being from a state school the probability is lower at 0.50. Suppose that when we break down the figures for men and women we have the following figures.
It may now be seen that now for both men and women the conditional probability of being admitted is higher for state school applicants, at 0.41 and 0.77, respectively.