1. You probability wonder what we’re going to do next!

2. Probability Basics Experiment Sample space Event
an activity with observable results or outcomes
Sample space
the set of all possible outcomes for an experiment
Event
any subset of the sample space

3. Probability Basics
where P(E) means the probability of an event occurring, n(E) means the number of individual outcomes in the event, and n(S) means the number of individual outcomes in the sample space.

4. Flip a coin
A well-known statistician named Karl Pearson once flipped a coin 24,000 times and recorded _______ “heads”; this result was extremely close to the theoretical probability and expected number of heads.
P(H) = _____ E(H) = _________

5. Spinners A B C A B C D
Spin each spinner once. Find P(A).

6. Spinners
If S = {1, 2, 3, 4, 5, . . ., 22, 23, 24}, find the probability of a:
Prime number
Even number
Number less than 10
Number less than 3 or greater than 17
Number less than 12 and greater than 9

7. Dice Roll a single die once. Find the following probabilities:
P(number greater than 4 or less than 2)
P(odd or even)
P(greater than 10)
P(at least 3)

8. Odds Odds for an event = P(for an event)/P(against the event)
If the odds for a horse winning a race are given, find the probability that the horse wins the race.
(a) 2:5 (b) 5:1 (c) 1:30

9. Experimental Probabilities
Conduct a poll to determine the probability:
That a Gordon transfer student drives a truck.
That a Gordon student has enjoyed a meal at the Dining Hall.
That a Gordon student has visited the BCM.

10. More vocabulary Complementary events
Everything else in the sample space
Examples:
If A = rolling a 1 or a 2 on a die, “A complement” is rolling
a 3, 4, 5, or 6
If R = it rains today, = it doesn’t rain today

11. Cards
Find the probability of drawing an ace from a standard deck of playing cards.
Find P(face card)
Find P(card with a value between 4 and 9)

12. More vocabulary Mutually exclusive (disjoint) events
When one event occurs, the other cannot possibly occur
Examples:
If A = even # on the roll of a die and B = 3 or 5,

13. P(A or B)
Mutually exclusive events
Non-mutually exclusive events

14. P(A or B)
Draw a card out of a standard 52-card deck. Find the probability that the
card is either black or an ace.
Roll a die once. If A = “even number on the die” and B = “rolling a 5 or 6”, find P(A or B).

15. Fundamental Counting Principle
If event M can occur in m ways and after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m x n ways.
(A tree diagram helps!)

16. Fundamental Counting Principle
How many outcomes are there for flipping 3 coins?
How many outcomes are there for rolling 2 dice? 3 dice?
If an ice cream shop has 32 flavors from which to choose and 7 toppings, how many different possibilities can I choose?

17. Fundamental Counting Principle
If automobile license plates consist of
4 letters followed by 3 digits, how many different license plates are possible if letters and digits may be repeated?

18. Multi-Stage Experiments
For any multi-stage experiment, the probability of the outcome along any path of a tree diagram is equal to the product of the probabilities along the path.

19. Problem
If the chance for success on the first stage of a rocket firing procedure is 96%, the second stage is 98%, and the final stage is 99%, find the probability of success on all three stages of the rocket firing procedure.

20. Flipping coins
List the sample space for two coins. (Use a tree diagram.)
Find the probability of at least one head.
List the sample space for three coins. (Use a tree diagram.)
Find the probability of exactly two heads.

21. Rolling Two Dice List the sample space.
Find the probability of a 3 on the first and a 3 on the second.
Find the probability of a sum of 7.
Find the probability of a sum of 10 or more.
Find the probability that both numbers are even.

22. Rolling Two Dice

23. Independent events
When the outcome of one event has no influence on the outcome of a second event, the events are independent.
For any independent events A and B, P(A and B) = P(A) x P(B).

24. Draw a ball from a container, replace it, and then draw a second ball.
Find the probability of a red, then a red.
Find P(no ball is red).
Find P(at least one red).
Find P(same color).

25. Dependent events
When the outcome of one event has an influence on the outcome of a second event, the events are dependent.

26. Draw a ball from a container, don’t replace it, and then draw a second ball.
Find the probability of a red, then a blue.
Find P(no ball is red).
Find P(same color).

27. Dependent vs. Independent events
Consider a bag that contains 219 coins of which 6 are rare Indian pennies. For the given pair of events A and​ B, complete parts​ (a) and​ (b) below.  
A: When one of the coins is randomly​ selected, it is one of the Indian pennies.
B: When another one of the coins is randomly​ selected, it is also one of the Indian pennies.
a. Determine whether events A and B are independent or dependent.​ (If two events are technically dependent but can be treated as if they are independent according to the​ 5% guideline, consider them to be​ independent.)
b. Find​ P(A and​ B), the probability that events A and B both occur.

28. A bag contains the letters of the word “probability”.
Draw 4 letters, one by one, from the bag. Find the probability of picking the letters of the word “baby” if the letters are drawn
With replacement
Without replacement

29. Consider the following 2 containers.
If a container above is selected at random, and then a letter is selected at random from
the chosen container, what is the probability that the letter chosen is an M?

30. For a challenge!
The Prisoner Problem
The Birthday Problem

31. Geometric Probabilities
If a dart hits the target below, find the probability that it hits somewhere in region 1. 2 1 1 2 3 4
The radius of the inner circle is 2 units and the radius of the outer circle is 4 units.

32. Games
A game is played by drawing 3 cards and replacing the card each time. A player wins if at least one face card is drawn. Find the probability of winning this game.
Find the probability of a “Yahtzee” in one throw of 5 dice.

33. Using Simulations Flipping a coin Rolling a die
Find the probability of a married couple having 2 baby girls.

34. Conditional Probabilities
When the sample space of an experiment is affected by additional info

35. Conditional Probabilities
If A = “getting a tail on the 1st toss of a coin” and B = “getting a tail on all 3 tosses of a coin”, find P(B|A).
What is the probability of rolling a 6 on a fair die if you know you rolled an even number?

36. Conditional Probabilities
With an auto insurance company, 60% of its customers are considered low-risk, 30% are medium-risk, and 10% are high-risk. After a study, the company finds that during a 1-yr period, 1% of the low-risk drivers had an accident, 5% of the medium risk drivers had an accident, and 9% of the high risk drivers had an accident. If a driver is selected at random, find the probability that the driver will have had an accident during the year.

37. Factorial Notation
Compute:

38. Permutations
From n objects, choose r of them and arrange them in a definite order. The number of ways this can be done is

39. Correspondences How many different ways can 4 swimmers
(Al, Betty, Carole, and
Dan) be placed in 4
lanes for a swim meet?

40. Permutations
If there are 12 players on a little league baseball team, how many ways can the coach arrange batting orders, with 9 positions on the field and at bat?

41. Permutations
How many arrangements are possible with the letters in the word:
algebra?
statistics?

42. Combinations
From n objects, choose subsets of size r (order unimportant). The number of ways this can be done is

43. Combinations
With 9 club members, how many different committees of 4 can be selected to attend a conference?
Braille Activity

44. Permutations & Combinations
There are 10 players on a U-6 soccer team, and the coach picks 5 starters. How many different groups of starters can the coach choose?
There are 10 members of a club. How many different “slates” could the membership elect as president, vice-president, and secretary/ treasurer (3 offices)?

45. Probability (with permutations/combinations)
Given a class of 12 girls and 9 boys,
in how many ways can a committee of 5 be chosen?
in how many ways can a committee of 3 girls and 2 boys be chosen?
What is the probability that a committee of 5, chosen at random, consists of 3 girls and 2 boys?