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In Sample Space, Experimental Outcomes, Events, Probabilities, we discussed various definitions. Here are some examples.
After various primaries there are four major candidates who could become president. Voting takes place in half a year. Over this year, many events could happen that would increase or decrease the support for candidates. After voting, the outcome is certain. But from today’s perspective, when voting is one year from now, outcomes are uncertain.
In Throw an Unfair Four Sided Dice, we show what happens when we throw an unfair four sided dice many times. In the example below, we will throw an unfair dice, where the probability of landing on the side with 1 is 50 percent, and the chance of landing on each successive side is half of the chance of landing on the previous side. This is a dice weighted towards the smaller numbers.
See the table below for the true probabilities of the unfiar dice.
# Load Library
library(tidyverse)
# Define a List of Experimental Outcomes
experimental.outcomes.list <- c(1,2,3,4,5,6)
# Probabilities on experimental outcomes
# Suppose dice is weighted towards 1
fracbase <- 0.50
experimental.outcome.prob <- c((1-fracbase)^0*fracbase,
(1-fracbase)^1*fracbase,
(1-fracbase)^2*fracbase,
(1-fracbase)^3*fracbase,
(1-fracbase)^4*fracbase,
(1-fracbase)^5)
# Show these in a Tibble
dice.true.prob <- tibble(dice.outcomes.list = experimental.outcomes.list,
dice.true.prob = experimental.outcome.prob)
kable(dice.true.prob) %>% kable_styling_fc()| dice.outcomes.list | dice.true.prob |
|---|---|
| 1 | 0.50000 |
| 2 | 0.25000 |
| 3 | 0.12500 |
| 4 | 0.06250 |
| 5 | 0.03125 |
| 6 | 0.03125 |
We throw the dice 10 times, each time, get one of the six possible experimental outcomes, the chance of getting these outcomes are determined by the true probabilities of the unfair dice. If we throw 10 times, that does not mean we will get exact side 1 five times. Side 1 having proability of 50 percent means for each throw there is a 50 percent chance of getting side 1. But each throw is independent of other throws, so we could just by chance end up with no side 1 after ten throws, although the chance of that happening is low. How many times did we get each side in the ten random tosses below?
# What could happen tomorrow?
# We live in a probabilistic world, drawing future from a hat
# If we draw 20 times, what happens in the future?
# It's pretty amazing, we get to see the future!
number.of.futures.to.draw = 10
future.dice.draws <- sample(experimental.outcomes.list,
size = number.of.futures.to.draw,
replace = TRUE,
prob = experimental.outcome.prob)
# A little tibble to show results
kable(tibble(which.future.dice = paste0('dice draws:', 1:number.of.futures.to.draw),
dice.draws = future.dice.draws)) %>% kable_styling_fc()| which.future.dice | dice.draws |
|---|---|
| dice draws:1 | 2 |
| dice draws:2 | 1 |
| dice draws:3 | 1 |
| dice draws:4 | 6 |
| dice draws:5 | 4 |
| dice draws:6 | 2 |
| dice draws:7 | 1 |
| dice draws:8 | 4 |
| dice draws:9 | 2 |
| dice draws:10 | 1 |
Following our definition for sample space above, we can actually define an alternative sample spaces, for example:
This is also a sample space with two sample points (or experimental outcomes). Following our definitions earlier, these two experimental outcomes are mutually exclusive and jointly exhaustive. We could also call these two experimental outcomes here two different events of the sample space with the six sample points.
Following our definition for sample space above, we can actually define an alternative sample spaces using these events:
This is also a sample space with three sample points (or experimental outcomes). Following our definitions earlier, these three experimental outcomes are mutually exclusive and jointly exhaustive. We could also call these three experimental outcomes here three different events of the sample space with the four paths of game play experimental outcomes (sample points).