## Probabilities in statistics.

Probability theory is a branch of mathematics concerned with the analysis of random phenomena.

Probability is the measure of the likelihood that an event will occur in a Random Experiment. A random experiment is a physical situation whose outcome cannot be predicted until it is observed.

Probability is quantified as a number between 0 and 1, where, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

## Calculating probabilities for a number of events:

```
import matplotlib.pyplot as plt
import numpy as np
## the basic formula
# counts of the different events
c = np.array([ 1, 2, 4, 3 ])
# convert to probability (%)
prob = 100*c / np.sum(c)
print(prob)
```

OUT: [10. 20. 40. 30.]

## Calculating probabilities for drawing marbles from a jar:

The following Python code shows probabilities and proportions calculation for case of drawing marbles of different colors - blue, yellow and orange - out of the box. It shows difference between probability and proportion and how far can it be depending the number of draws.

```
# colored marble counts
blue = 40
yellow = 30
orange = 20
totalMarbs = blue + yellow + orange
# put them all in a jar
jar = np.hstack((1*np.ones(blue),2*np.ones(yellow),3*np.ones(orange)))
# now we draw 500 marbles (with replacement)
numDraws = 500
drawColors = np.zeros(numDraws)
for drawi in range(numDraws):
# generate a random integer to draw
randmarble = int(np.random.rand()*len(jar))
# store the color of that marble
drawColors[drawi] = jar[randmarble]
# now we need to know the proportion of colors drawn
propBlue = sum(drawColors==1) / numDraws
propYell = sum(drawColors==2) / numDraws
propOran = sum(drawColors==3) / numDraws
# plot those against the theoretical probability
plt.bar([1,2,3],[ propBlue, propYell, propOran ],label='Proportion')
plt.plot([0.5, 1.5],[blue/totalMarbs, blue/totalMarbs],'b',linewidth=3,label='Probability')
plt.plot([1.5, 2.5],[yellow/totalMarbs,yellow/totalMarbs],'b',linewidth=3)
plt.plot([2.5, 3.5],[orange/totalMarbs,orange/totalMarbs],'b',linewidth=3)
plt.xticks([1,2,3],labels=('Blue','Yellow','Orange'))
plt.xlabel('Marble color')
plt.ylabel('Proportion/probability')
plt.legend()
plt.show()
```