Stat 20: Intro to Probability and Statistics Lecture 12: More Probability Tessa L. Childers-Day UC Berkeley 10 July 2014
By the end of this lecture... You will be able to: Use the theory of equally likely outcomes to carefully determine the probability of a given event Determine whether two events are mutually exclusive Apply the addition rule Calculate probabilities of more complicated events 2 / 22
Three theories of probability We think about the situation, and use the appropriate theory Subjective Equally Likely Outcomes Frequency 3 / 22
Properties of Probability Some things that must be true: Probabilities are between 0% and 100% (or 0 and 1) The probabilities of all possible events add to 100% (or 1) P(something) = 100% P(opposite thing) = 1 - P(opposite thing) 4 / 22
Probability Basics Strategies and formulas for finding probabilities: Draw a box model, fill it with tickets, draw randomly Conditional probability P(A B) Independence and dependence Multiplication Rule: P(A and B) = P(B) P(A B) = P(A) P(B A) OR P(A and B) = P(A) P(B) 5 / 22
Counting the Ways It is always an option to list all possible outcomes, see how many match up chance = number of outcomes that match desired event total number of outcomes Warning: All possibilities must be listed, not just all combinations. 6 / 22
Counting the Ways (cont.) Example: Suppose we are flipping a coin 4 times. What is the probability of getting 2 heads and 2 tails? List all possibilities: All 4 Heads 3 Heads, 1 Tail 2 Heads, 2 Tails 1 Head, 3 Tails All 4 Tails 7 / 22
Counting the Ways (cont.) Example: Suppose we are flipping a coin 4 times. What is the probability of getting 2 heads and 2 tails? List all possibilities: All 4 Heads: HHHH (1 way) 3 Heads, 1 Tail: HHHT, HHTH, HTHH, THHH (4 ways) 2 Heads, 2 Tails: HHTT, HTTH, TTHH, THHT, HTHT, THTH (6 ways) 1 Head, 3 Tails: HTTT, THTT, TTHT, TTTH (4 ways) All 4 Tails: TTTT (1 way) 8 / 22
Example: Counting Cards Find the probability that a single card drawn from a standard deck is a king or a club. Let s list the ways this could happen: There are 4 kings in the deck There are 13 clubs in the deck There are 52 cards in the deck What is P(K or )? 9 / 22
The Addition Rule To find the chance of either of two events occurring, add the chance of the 1 st to the chance of the 2 nd, and subtract the chance of both events occurring: P(A or B) = P(A) + P(B) - P(A and B) Note that P(A or B) P(A) and any other parts 10 / 22
The Addition Rule (cont.) P(A or B) = P(A) + P(B) - P(A and B) Can use this rule to explain our earlier example: What is the probability that a single card drawn from a standard deck is a king or a club? P(K or ) = P(K) + P( ) - P(K and ) = 4 52 + 13 52 1 52 = 16 52 = 0.308 11 / 22
Mutual Exclusivity 2 events are mutually exclusive if one event occurring excludes the other event from occurring. P(A and B) = 0. 2 events are not mutually exclusive if one event occurring does not exclude the other event from occurring. P(A and B) 0. 12 / 22
Mutual Exclusivity (cont.) Mutual exclusivity and independence are NOT the same thing! Independent: P(A B) = P(A) Mutually exclusive: P(A and B) = 0 13 / 22
Mutual Exclusivity (cont.) Mutually Exclusive: P(A and B) = 0 This affects the addition rule: P(A or B) = P(A) + P(B) - P(A and B) P(A or B) = P(A) + P(B), if A and B are mutually exclusive 14 / 22
Mutual Exclusivity (cont.) P(A or B) = P(A) + P(B), if A and B are mutually exclusive Can use this rule to explain our earlier example: Suppose we are flipping a coin 4 times. What is the probability of getting 2 heads and 2 tails? P(2H 2T) = P(HHTT or HTTH or TTHH or THHT or HTHT or THTH) = P(HHTT) + P(HTTH) + P(TTHH) + P(THHT) + P(HTHT) + P(THTH) ( ) 1 4 ( ) 1 4 ( ) 1 4 ( ) 1 4 ( ) 1 4 = + + + + + 2 2 2 2 2 ( ) 1 4 = 6 = 0.375 2 ( ) 1 4 2 15 / 22
Summary of Rules The following rules can be applied in calculating probabilities Complement Rule: P(A) = 1 - P(Not A) Multiplication Rule: P(A and B) = P(A) P(B A) [and a special case when A and B are independent] Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) [and a special case when A and B are mutually exclusive] Combining these rules allows us to calculate many different probabilities 16 / 22
Examples Are the following events mutually exclusive? 1 Select a student in your class, and he/she has blond hair and blue eyes 2 Select a student in your college, and he/she is a sophomore and a Chemistry major 3 Select any course in your college, and it is a calculus course and an English course 4 Select a registered voter, and he/she is a Republican and a Democrat When rolling a die once you get: 5 An even number, and a number less than 3 6 A prime number, and an odd number 7 A number greater than 3, and a number less than 3 17 / 22
Examples (cont.) A single card is drawn from a deck. Find the probability of selecting the following: 1 A 4 or a diamond 2 A club or a diamond 3 A jack or a black card 18 / 22
Examples (cont.) Three dice are thrown at once. Find the chance that 1 All three dice show 4 spots 2 The third die shows 4 spots, given the first two show 4 spots 3 All three dice show the same number of spots 4 Two or fewer dice show 4 spots 5 The sum of the spots is 5 6 At least one 5 is rolled 19 / 22
Examples (cont.) In an upper division statistics class there are 18 juniors and 10 seniors. 6 seniors are females, 12 juniors are males. If a student is selected at random, find the chance of selecting the following: 1 A junior or a female 2 A senior or a female 3 A junior or a senior 20 / 22
Examples (cont.) An urn contains 6 red balls, 2 green balls, and 2 white balls. Find the chance of selecting the following: 1 In one draw, a red or white ball 2 In two draws, with replacement, 2 red balls or 2 white balls 3 In draws, without replacement, 2 red balls or 2 white balls 21 / 22
Important Takeaways When in doubt, count it out Be careful not to double count Addition rule Mutually exclusive vs. non-mutually exclusive events Next time: Calculating probabilities for independent, binary events 22 / 22