Probability Mass Function
Last time, we developed the probability mass function for the binomial distribution. The probability of choosing \(k\) observations among a sample size of \(n\), each observation with prior probability \(p\), is given by
\[P(k) = \binom{n}{k}p^{k}(1-p)^{n-k}\]
Note the usual properties of probability:
- each probability is between zero and one (inclusively)
\[0 \leq P(k) \leq 1 \quad\text{for each } k\] - all probabilities add up to 100%
\[1 = \displaystyle\sum_{k = 0}^{n} \binom{n}{k}p^{k}(1-p)^{n-k}\]
From One to Many
There are 4 parking spaces in front of the boba place. Suppose that each parking space tends to be occupied about 57 percent of the time. What is the probability that exactly 3 of the parking spaces are open?
\[k \in \{0, 1, 2, 3, 4\}\]
There are 4 parking spaces in front of the boba place. Suppose that each parking space tends to be occupied about 57 percent of the time. What is the probability that at most 2 of the parking spaces are open?
\[k \in \{0, 1, 2, 3, 4\}\]
There are 32 parking spaces in a row in a UC Merced parking lot. Suppose that each parking space tends to be occupied about 81 percent of the time. What is the probability that exactly 4 of the parking spaces are open?
\[k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\] \[\cup \{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32\}\]
There are 32 parking spaces in a row in a UC Merced parking lot. Suppose that each parking space tends to be occupied about 81 percent of the time. What is the probability that more than 5 of the parking spaces are open?
\[k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\] \[\cup \{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32\}\]
Leveraging Complements
There are 32 parking spaces in a row in a UC Merced parking lot. Suppose that each parking space tends to be occupied about 97 percent of the time. What is the probability that at least one of the parking spaces is open?
\[k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}\] \[\cup \{17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32\}\]
Looking Ahead
- due Fri., Feb. 10:
- WHW4
- JHW2
- Demographics Survey Part 1
- Be mindful of before-lecture quizzes
No lecture session for Math 32:
- Feb 20, Mar 10, Mar 24
Exam 1 will be on Wed., Mar. 1
- more information in weekly announcements
