Probability Definitions
Key Concepts of Probability
Probability Math Problems
More probability math!
Chapter 6: Probability Distributions
100

What is a Simulation?

What is: a technique used to recreate a random event. The goal in simulations is to measure how often a goal is observed.

100

Impossible Event

P = 0

100

  1. A local bakery tracked their pie sales for a week. Out of 200 pies sold, 80 were apple, 60 were cherry, 40 were blueberry, and 20 were peach. Using the empirical approach, what is the probability that a randomly selected pie is:

a) Apple

b) Cherry

c) Either blueberry or peach

Bakery pie sales:
a) P(Apple) = 80/200 = 0.40 or 40%
b) P(Cherry) = 60/200 = 0.30 or 30%
c) P(Blueberry or Peach) = (40 + 20)/200 = 60/200 = 0.30 or 30%

100

The probability of rain on Saturday is 0.3, and the probability of rain on Sunday is 0.4. Assuming the weather on these days is independent, what is the probability that it will rain on both days?



  1. P(Rain Saturday) = 0.3

P(Rain Sunday) = 0.4

P(Rain both days) = 0.3 × 0.4 = 0.12 or 12%



100

What is a Discrete Random Variable

Has either a finite or countable number of values 

200

What is the Empirical Approach? 

The probability of an event E is approximately the number of times event E is ovserved divided by the number of repetitions of the experiment. 

200

More likely events

 P > .50

200

  1. In a class of 30 students, a survey about favorite sports revealed the following

    • 12 prefer football

    • 8 prefer basketball

    • 6 prefer soccer

4 prefer tennis

What is the empirical probability that a randomly chosen student from this class prefers:

a) Football

b) Basketball or soccer

c) A sport other than tennis

Class sports preferences:
a) P(Football) = 12/30 = 0.40 or 40%
b) P(Basketball or soccer) = (8 + 6)/30 = 14/30 ≈ 0.47 or 47%
c) P(Sport other than tennis) = (12 + 8 + 6)/30 = 26/30 ≈ 0.87 or 87%

200

A fair six-sided die is rolled twice. What is the probability of rolling an even number on the first roll and a number greater than 4 on the second roll?



  1. P(Even) = 3/6 = 1/2

P(>4) = 2/6 = 1/3

P(Even then >4) = 1/2 × 1/3 = 1/6 ≈ 0.1667 or about 16.67%



200

What is a continuous random variable

infinitely many values

300

What is a random process?

What is: scenarios where the outcome of any particlaar trial of an experiment is unknown, but the proportion of a particular outcome approaches a specific value as the # of trials increases

300

Less likely events

P < 0.50

300

  1. In a bag of 500 marbles, a child counted

    • 150 red marbles

    • 125 blue marbles

    • 100 green marbles

    • 75 yellow marbles

    • 50 white marbles
      Using the empirical approach, calculate the probability of randomly drawing:
      a) A blue marble
      b) A marble that is not green
      c) Either a red or yellow marble

Bag of marbles:
a) P(Blue marble) = 125/500 = 0.25 or 25%
b) P(Not green) = (150 + 125 + 75 + 50)/500 = 400/500 = 0.80 or 80%
c) P(Red or yellow) = (150 + 75)/500 = 225/500 = 0.45 or 45%

300

 A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn with replacement. What is the probability of drawing a red marble followed by a blue marble?

  1. P(Red) = 5/8

P(Blue) = 3/8

P(Red then Blue) = 5/8 × 3/8 = 15/64 ≈ 0.2344 or about 23.44%



300

Please write the standard deviation of a discrete random variable X formula 

σ = √[Σ (x - μ)² p(x)]


400

What is Probability?

What is: a likelihood of a random phenomenon or chance behavior occuring 

400

Certain Event

P = 1

400

Consider drawing a single card from a standard 52-card deck. Let's define two events:

Event A: Drawing a face card (Jack, Queen, or King)

Event B: Drawing a number card (2 through 10)

Calculate the probability of either Event A or Event B occurring.


  1. First, let's identify the outcomes for each event:

    • Event A (face cards): 12 cards (3 face cards in each of 4 suits)

    • Event B (number cards): 36 cards (9 number cards in each of 4 suits)

  2. Verify that these events are disjoint:
    Indeed, they are disjoint because a card cannot be both a face card and a number card simultaneously.

  3. For disjoint events, we use the addition rule:
    P(A or B) = P(A) + P(B)

  4. Calculate P(A):
    P(A) = 12/52 = 3/13

  5. Calculate P(B):
    P(B) = 36/52 = 9/13

Apply the addition rule:

P(A or B) = P(A) + P(B) = 3/13 + 9/13 = 12/13 ≈ 0.9231

400

A local book club has a shelf with 15 different novels. For their next meeting, they want to select 4 books for discussion. The order of selection doesn't matter, as all selected books will be discussed equally. How many different combinations of 4 books can be selected from the 15 novels?

  1. This is a straightforward combination problem. We need to calculate C(15,4).

C(15,4) = 15! / (4! × (15-4)!) = 15! / (4! × 11!)

= (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1)

= 1365



400

What are the requirements of a binomial experiment? 

1. the experiment is performed a fixed # of times

2. The trials are independent 

3. For each trial, there are two mutually exclusive outcomes

4. The probability of success ist he same for each trial 

500

What is the law of large numbers?

As the # of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.
500

Equally Likely Outcome 

each outcome has the same probability of occurring 

500


  1. A local ice cream shop wants to determine the probability of a customer choosing vanilla ice cream. The shop owner decides to use the empirical approach by observing customer choices over a week. During this period, 420 customers bought ice cream cones. Of these, 126 customers chose vanilla.
    - What is the empirical probability that a customer will choose vanilla ice cream? If 1000 customers visit the shop next week, how many would you expect to choose vanilla ice cream based on this empirical probability?

  1. P(Vanilla) = Number of customers who chose vanilla / Total number of customers
    P(Vanilla) = 126 / 420 = 0.3 or 30%

Expected number of customers choosing vanilla out of 1000:

Expected number = Probability × Total customers

    Expected number = 0.3 × 1000 = 300 customers

500

A local charity organization is forming a committee to oversee their annual fundraising event. They have 15 volunteers available, and they need to select 6 people to form the committee. How many different ways can they form this committee?

  1. Total number of volunteers (n) = 15Number of people needed for the committee (r) = 6
  2. C(n,r) = n! / (r! * (n-r)!)
    Where n! represents the factorial of n.
  3. Calculate:
    15! = 1,307,674,368,000
    6! = 720
    9! = 362,880
  4. C(15,6) = 1,307,674,368,000 / (720 * 362,880)
    = 1,307,674,368,000 / 261,273,600
    = 5,005
500

A software company releases a new app. Based on previous data, they estimate that 75% of users who download the app will continue using it after the first week. If 20 people download the app, what is the probability that exactly 15 of them will continue using it after the first week?

  1. Identify the components of the binomial distribution:
    n = 20 (number of trials/people who download the app)
    p = 0.75 (probability of success, i.e., continuing to use the app)
    1 - p = 0.25 (probability of failure)
    x = 15 (number of successes we're interested in)
  2. Recall the binomial probability formula:
    P(X = x) = C(n,x) * p^x * (1-p)^(n-x)
    Where C(n,x) is the number of combinations of n items taken x at a time.
  3. Calculate C(n,x):
    C(20,15) = 20! / (15! * (20-15)!)
    = 20! / (15! * 5!)
    = 15,504
  4. Apply the formula:
    P(X = 15) = 15,504 * (0.75)^15 * (0.25)^(20-15)
    = 15,504 * (0.75)^15 * (0.25)^5
  5. Calculate:
    (0.75)^15 ≈ 0.0134
    (0.25)^5 ≈ 0.0009765625P(X = 15) = 15,504 * 0.0134 * 0.0009765625
    ≈ 0.2032 (rounded to 4 decimal places)

Therefore, the probability that exactly 15 out of 20 people will continue using the app after the first week is approximately 0.2032 or 20.32%.

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