Skill 20: Calculating Probabilities

What is probability?

Probability measures how likely it is for an event to occur.

What is probability?

h𝑡 𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦=¿𝑤𝑎𝑦𝑠 𝑦𝑜𝑢𝑟 𝑒𝑣𝑒𝑛𝑡𝑐𝑎𝑛𝑜𝑐𝑐𝑢𝑟

𝑡𝑜𝑡𝑎𝑙¿𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠¿

Calculating Theoretical Probabilities

Example #1: A group of five cards are numbered 1 – 5. You choose one card at random. Find each theoretical probability.a) b)

Calculating Theoretical Probabilities

Example #1: A group of five cards are numbered 1 – 5. You choose one card at random. Find each theoretical probability.c) d)

Calculating Theoretical Probabilities

Example #2: A bucket contains 15 blue pens, 35 black pens, and 40 red pens. You pick one pen at random. Find each theoretical probability.a) b)

Calculating Theoretical Probabilities

Example #2: A bucket contains 15 blue pens, 35 black pens, and 40 red pens. You pick one pen at random. Find each theoretical probability.c) d)

You Try:

A jar contains 30 red marbles, 50 blue marbles, and 20 white marbles. You pick one marble from the jar at random. Find each theoretical probability.a) b) c) d)

Calculating Theoretical Probabilities

Sometimes, certain outcomes can belong in more than one event. Be sure to not count these outcomes twice!

Here’s an example to demonstrate…

Calculating Theoretical Probabilities

Example #3: Ten slips of paper are placed into a container. Each is labeled with a number from 1 through 10. Find each probability of picking each of the following from the container.a) b)

Calculating Theoretical Probabilities

Example #3: Ten slips of paper are placed into a container. Each is labeled with a number from 1 through 10. Find each probability of picking each of the following from the container.c)

Calculating Theoretical Probabilities

Example #4: You randomly choose a number between 1 and 10. What is the probability that you choose a multiple of 2 or 3?

You Try:

Calculating Theoretical Probabilities by Using Combinations and PermutationsExample #5: What is the theoretical probability of being dealt exactly two 7’s in a 5-card hand from a standard 52-card deck?

Calculating Theoretical Probabilities by Using Combinations and PermutationsExample #6: There are 225 juniors and 255 seniors at your school. The school chooses 5 juniors and seniors as Student All-Stars. What is the theoretical probability that exactly 3 of the Student All-Stars will be juniors?

You Try:

1. Francis randomly selects two pieces of fruit from a basket containing 8 oranges and 4 apples. Find the probability that she selects exactly 1 orange and 1 apple.

2. A group of 30 students from your school is part of the audience for a TV game show. The total number of people in the audience is 150. What is the theoretical probability of exactly 3 students from your school being selected as contestants out of 9 possible contestant spots?

Calculating Theoretical Probabilities GeometricallyExample #7: The rectangular yard shown below has a circular pool and a triangular garden. The ball from an adjacent golf course lands at a random point within the yard. Find each theoretical probability.a) The ball lands in the pool.

Calculating Theoretical Probabilities Geometricallyb) The ball lands in the garden.

c) The ball lands in the garden or the pool.

Calculating Theoretical Probabilities Geometricallyd) The ball does not land in the pool.

You Try:

The total number of degrees in a circle is 360°. Use the spinner to find the theoretical probability of each event. 1. The pointer landing in D2. The pointer landing in A or B3. The pointer NOT landing in C

Who Uses This? Actuaries!

What do actuaries do?- Actuaries evaluate the likelihood of future events—using numbers, not crystal balls! Insurance companies, financial institutions, and the government are top employers.How do I become an actuary?- Major in Actuarial Science, and then pass a series of exams (you can “level up”, like in karate). The more exams you pass, the more prestige (and $) you have.

Who Uses This? Actuaries!

How much money could I make?

Exit Quiz

1. What is the theoretical probability of getting a sum of 7 on one roll of two fair six-sided dice?2. What is the theoretical probability of being dealt exactly one ace in a 5-card hand from a standard 52-card deck?3. A rectangular landing pad measures 20’ x 25’ with a rectangular target on the pad measuring 6’ x 8’. What is the probability that a model rocket falling randomly on the pad will land in the target?