Once upon a time, there was a young girl named Goldilocks. She was bored at home so she thought about going for a walk.
Great idea. However, there is a 35% chance of rain all day. Along with 35% chance of rain comes a 15% chance of lightning if it rains.
I feel like there's an 80% chance that I will go on a walk today.
Goldilocks and the Three Bears
Going for a walk when it's raining out is independent while the chance of lightning is a dependent event. Therefore, I can do some calculations.
Walking in the RainEvent A: Going for a walkEvent B: It rainsP(A and B) = P(A) x P(B)=(0.8 x 0.35) 100=0.28 x 100=28%
Rain and LightningEvent A: It rainsEvent B: Lightning strike*Lighting is conditional*P(A)= 35% or 0.35P(B|A)= 15% or 0.15P(A and B) = P(A) x P(B|A)=(0.35 x 0.15) 100=0.05 x 100=5%
Because of the low chance of rain and lightning, she decided to go on her walk.
1 2 3 4 5 6 7 8 9
1 4 10 20 35 56 84 120 165
1 3 6 10 15 21 28 36 45
1 5 15 35 70 126 210 330 495
G 1 1 1 1 1 1 1 1
This porridge is way too hot!
Based on her findings from her knowledge of Pascal's triangle, Goldilocks found that through the entire forest she could take up to 495 different paths to get to her destination. She knew that her total number of routes was correct since she knew that row 12 number 8 or 12C8 was 495.
As Goldilocks drew closer and closer to the forest she noticed that the forest was laid out into a grid-like pattern. Knowing that she could only walk east and south, she remembered that she could use Pascal's triangle to determine how many ways she could get from her location to her destination.
Nobody's answering the door, I suppose I'll just go inside.
This porridge is way too cold!
As she walked along her path, she came across a house. She decided to see if anyone was home.
Ahhh, this porridge is just right!...but it could use some toppings!
When Goldilocks walks inside, the first thing she sees is 3 bowls of porridge. She decides she's going to test them.
Goldilocks went through the cupboards and found the following toppings: apple slices, blueberries, syrup, honey, vanilla, milk, and cream. These were grouped into 3 categories: Fruits, sauces, and dairy.
Honey
I wonder how many different ways I can top my porridge. There are 3 groups of toppings, or in other words, 3 stages to the decision making process, I think I'll use a tree diagram and the fundamental counting principle to figure it out!
Fruits A BSauces S H V S H VDairy M C M C M C M C M C M CTotal= 2 x 3 x 2 = 12 ways to top the porridge.
Honey
After tasting all 3 bowls of porrdige, she determines her experimental probability of enjoying the porridge was 1/3 or 33%. She also realized that there were events and outcomes in this experiment. The event was her tasting the porridge and the outcomes were her liking or disliking the porridge.
I also wonder what the theoretical probability is of randomly picking both fruits.
If there are 7 toppings in total, then my denominator is 7. If I want to pick the 2 fruits out of the 7 at random, my numerator would be 2.
Let A be picking both fruits.P(A)= 2/7 = 0.2857 = 29% (Approximately)
She then decided to head to the living room.
This chair is too big!
However, as soon as she sat down to rest the chair broke into pieces!
Goldilocks is very tired at this point and would like to go lay in bed so she wanders up the stairs towards the bedroom.
As she is going up to bed, she wonders how many beds out of the three she will like. She assumed that she would like as many beds as she did bowls of porridge and chairs. So, she decided to use the indirect method to determine the theoretical probability of her liking a bed.
This chair is too big too!
Ahhh this chair is just right!
This bed is too hard!
After completing her trials of the 3 chairs, she realized that she only liked 1 out of the 3 chairs. Therefore, she realized that her experimental probability of liking the chairs was 1/3 or 33%.
This bed is too soft!
Ahhh, this bed is just right!
P (Beds) = 1 - P (Disliking a chair) = 1 - 2/3 = 1 - 0.667 = 3/3 - 2/3 or 1 - 0.667 P (Beds) = 1/3 or 0.333 or 33%
Goldilocks remembers that she can not fall asleep without stuffed animals beside her. Luckily, the owners of the the home have tons of teddy bears in several different colours. She wonders how many different colour patterns she can make with the bears if she uses anywhere from 1-6 bears. She decides to use the additive counting principle to solve her problem.
1 bear: 6 = 62 bears: 6 x 5 = 303 bears: 6 x 5 x 4 = 1204 bears: 6 x 5 x 4 x 3 = 3605 bears: 6 x 5 x 4 x 3 x 2 = 7206 bears: 6 x 5 x 4 x 3 x 2 x 1 = 720Total colour patterns = 6 + 30 + 120 + 360 + 720 + 720 = 1956
Goldilocks notices that there are 4 different coloured pillows on the bed; one white, one black, one grey, and one sand. She wonders how many different ways she can order the pillows. To calculate this she knows to use permutations.
Someone's been eating my porridge!
Someone's been eating my porridge and they ate it all up!
Just then, the bears came home.
nPr = n! / (n - r)!4P4 = 4! / (4 - 4)!4P4 = 4! / 0!4P4 = 24 / 14P4 = 24Therefore, there are 24 ways that she can order the pillows.
Someone's been eating my porridge!
Dark Memory Foam
Someone's been sitting in my chair!
Next Goldilocks wondered which pillow she wanted to sleep with. So, she broke them into 2 categories: Dark colour vs. light colour and memory foam vs. feather. She wanted to find the probability that she would randomly pick a dark or a memory foam pillow. So, she created a Venn diagram and calculated the probability of the non-mutually exclusive events.
P (A or B) = P (A) + P ( B) - P (A and B)P (A or B) = 2/4 + 2/4 - 1/4P (A or B) = 4/4 - 1/4P (A or B) = 3/4 or 75%
Someone's been sitting in my chair and they've broken it to pieces!
Someone's been sitting in my chair!
Someone's been sleeping in my bed!
Someone's been sleeping in my bed and they're still there!
Finally, before she fell asleep she wanted to pick a 2 blanket combination to keep her warm. There were 6 different blankets to choose from. She decided to use her knowledge of combinations to calculate how many combinations she could choose from.
nCr = n! / (n-r)! r!10C2 = 10! / (10 - 2)! 2!10C2 = 3,628,800 / 80,64010C2 = 45Therefore, there are 45 different blanket combinations that she can choose from.
Someone's been sleeping in my bed!
Just then, Goldilocks woke up. She saw three bears. She screamed, "Help!"And she jumped up and ran out of the room.Goldilocks ran down the stairs, opened the door and ran away into the forest. She never returned to the home of the three bears.THE END
Image Attributions: (https://pixabay.com/en/apple-malus-kernobstgewaechs-large-336015/) - stux - License: Free for Commercial Use / No Attribution Required (https://creativecommons.org/publicdomain/zero/1.0)